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Uniqueness of Power Flow Solutions Using Graph-theoretic NotionsHaixiang Zhang, SangWoo Park, Javad Lavaei and Ross Baldick

Abstract—This paper extends the uniqueness theory in [1]and establishes general necessary and sufficient conditions forthe uniqueness of P -Θ power flow solutions in an AC powersystem using some properties of the monotone regime andthe power network topology. We show that the necessary andsufficient conditions can lead to tighter sufficient conditionsfor the uniqueness in several special cases. Our results arebased on the existing notion of maximal girth and our newnotion of maximal eye. Moreover, we develop a series-parallelreduction method and search-based algorithms for computingthe maximal eye and maximal girth, which are necessary forthe uniqueness analysis. Reduction to a single line using theproposed reduction method is guaranteed for 2-vertex-connectedSeries-Parallel graphs. The relations between the parametersof the network before and after reduction are obtained. It isverified on real-world networks that the computation of themaximal eye can be reduced to the analysis of a much smallerpower network, while the maximal girth is computed during thereduction process.

I. INTRODUCTION

The AC power flow problem plays a crucial role in var-ious aspects of power systems, e.g., the daily operationsin contingency analysis and security-constrained dispatch ofelectricity markets. In essence, the goal of the AC power flowproblem is to solve for the complex voltage of each bus thatdetermines the power system set-point. However, the presenceof sinusoidal functions in the AC power flow equations makesit difficult to analytically solve the equations, if not impossible.Moreover, the periodicity of sinusoidal functions destroys theuniqueness of the AC power flow solution, even when eithervoltage magnitudes or phase angle differences are limited tothe “physically realizable” regime [1], [2], [3], [4]. Hence,unexpected operating points may appear for some systemconditions and can jeopardize the normal operations of powersystems. Conditions that ensure the existence of a unique“physically realizable” power flow solution are important butnot fully understood.

For a special case of the AC power flow problem, theuniqueness property of the P -Θ power flow problem [5] hasbeen studied in [1]. In the P -Θ power flow problem, themagnitude of the complex voltage at each node is given andthe objective is to find a set of voltage phases such that thepower flow equations are satisfied. The “physically realizable”constraint requires that the angular difference across every

Haixaing Zhang is with the Department of Mathematics, University ofCalifornia Berkeley. Email: haixiang_zhang@berkeley.edu. Sang-Woo Park and Javad Lavaei are with the Department of Industrial Engi-neering and Operations Research, University of California Berkeley. Emails:{spark111, lavaei }@berkeley.edu. Ross Baldick is with the De-partment of Electrical and Computer Engineering, University of Texas atAustin. Email: baldick@ece.utexas.edu.

The authors affiliated with UC Berkeley were in part supported by grantsfrom ARO, ONR, AFOSR and NSF. Ross Baldick was in part supported bythe University of Texas Faculty Development Program.

line lies within the stability limit of π/2 for lossless net-works. Sufficient conditions (on the angular differences) thatdepend on the topological properties of the power networkare established in [1]. Specifically, the authors proposed thenotion of monotone regime and an upper bound on the angulardifferences based on the power network topology, whichtogether can ensure the uniqueness of solutions. However, dueto the nonlinear property of sinusoidal functions and the low-rank structure of angular differences, it is unclear to whatextent the sufficient conditions given in [1] are necessary.

The goal of this paper is to provide more general necessaryand sufficient conditions for the uniqueness, using the notionof maximal eye defined in Section III and the notion ofmaximal girth introduced in [1]. The paper also designsalgorithms to compute these graph-theoretic parameters.

A. Main results

In this paper, we extend the uniqueness theory of P -Θ power flow problem proposed in [1]. We focus on theuniqueness of the power flow problem in a stronger senseand derive general necessary and sufficient conditions thatdepend only on the choice of the monotone regime and networktopology. Under certain circumstances, the general conditionscan be simplified to obtain tighter sufficient conditions. Inaddition, some algorithms for computing the maximal eyeand the maximal girth of undirected graphs are proposed. Areduction method is designed to reduce the size of graphsand accelerate the computation process. More specifically, thecontributions of this paper are three-folds:• We extend the uniqueness theory of P -Θ problem to a

stronger sense. In [1], two solutions for cyclic graphsare are treated as different if all phase angle vectorsare different (this is referred to as weak uniqueness).In contrast, in this paper, we consider the uniqueness inthe usual sense, i.e., two solutions are different if theyare different in at least one phase angle (this is referredto as strong uniqueness). A constant called the maximaleye is developed to classify all network topology thatensure strong uniqueness. Numerical results show that themaximal eye gives more reasonable constraints comparedto its counterpart for weak uniqueness, which is knownas the maximal girth. Moreover, for 2-vertex-connectedSeries-Parallel graphs, we prove that the maximal eye isequal to the maximal girth.

• We propose general necessary and sufficient conditionsfor both the strong and the weak uniqueness. The con-ditions are derived by Farka’s Lemma, which gives thedual problem with simpler solutions. Sufficient conditionsfor the weak uniqueness in [1] and their counterpartsfor the strong uniqueness are derived directly from thegeneral conditions. In the special case when the power

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network is a single cycle or is lossless, stronger necessaryand sufficient conditions that do not contain sinusoidalfunctions are derived.

• Finally, we develop a reduction method, named theSSPR method, that can accelerate the computation of themaximal eye and the maximal girth. The SSPR methodis proved to reduce 2-vertex-connected Series-Parallelgraphs to a single line, independent of the choice ofthe slack bus. The relationship between the maximaleye (girth) of graphs before and after the reduction isunveiled. When applying the SSPR method to real-worldexamples, the maximal eye is usually not changed overthe reduction process, while the maximal girth is alreadycomputed during the reduction process. We also designsearch-based algorithms for computing the maximal eyeand the maximal girth, which are able to compute theexact value for graphs with up to 100 nodes beforereduction.

In summary, this paper constitutes a substantial generaliza-tion of the uniqueness theory in [1]. A stronger notion ofuniqueness is proposed and general necessary and sufficientconditions are proposed. These two combined provides atool for analyzing large-scale power networks and enables adeeper understanding of the uniqueness of the P -Θ power flowproblem.

B. Related work

The study of solutions to the power flow problem has a longhistory dating back to [6], which gave an example showingthe general non-uniqueness of solutions for the power flowproblem. Then, the number of solutions of the power flowproblem was estimated in [7], which also characterized thestability region for the power flow problem. However, theseearly works only considered lossless transmission networksconsisting of PV buses. Under the assumption that resistivelosses are negligible, conditions for the existence and unique-ness of both real power-phase (P -Θ) problem, and reactivepower-voltage (Q-V ) problem were derived in [8], [5].

In another line of work, the topology structure of the powernetwork was also considered to derive stronger conditions forthe uniqueness. The number of solutions was estimated forradial networks in [2], [9], and later for general networks.Moreover, a more recent work [10] gave several algorithmsto compute the unique high-voltage solution. In this paper,we consider the P -Θ problem [5] for general lossy powernetworks and utilize the topology information. We refer to [1]for a more detailed review of the existing literature.

The fixed-point technique is often used for proving theexistence and uniqueness of equations. For the power flowproblem, the fixed-point technique was first utilized in [11]and was further developed by several works [12], [13], [14],[15], [16], [17]. Another more recently applied approach is totreat the P -Θ power flow problem as a rank-1 matrix sensingproblem and solve its convex relaxation counterpart [18],[19]. The work [20] also considered the domain of voltagesover which the power flow operator is monotone. However,the relation between the rank-1-constrained problem and itsconvexification is not clear for general power networks.

The work [21] presented a unifying framework for networkproblems on the n-torus. The framework applies to the ACpower flow problem when the power networks are lossless.The idea of considering the regime when the power flow oneach line is monotone was extended to lossy power networksin [1]. The regime where the power flow on a line increasesmonotonically with the angle difference across the line – calledthe monotone regime in this paper – was proposed. In [1], itwas also shown that the solution of P -Θ problem is uniqueunder the assumption that angle differences across the linesare bounded by some limit related to the maximal girth of thenetwork [22].

The existing algorithms in the literature cannot be directlyused to compute maximal eye (introduced in Section III) ormaximal girth. A related problem is computing the maximalchordless cycle as an upper bound to these parameters. Thecomputation of maximal chordless cycles was proved to beNP-complete in [23]. Efficient algorithms for enumeratingchordless cycles were proposed in [24], [25] and both takelinear time to enumerate a single chordless cycle. The algo-rithms for enumerating maximal chordless cycles can be easilymodified to compute the minimal chordless cycle containing agiven edge. Series-parallel reduction method was introduced asan alternative definition of Generalized Series-Parallel (GSP)graphs in [26]. Under the assumption that the slack bus is thelast bus to be reduced, all GSP graphs can be reduced to asingle line [1]. However, whether the series-parallel reductionmethod can still reduce GSP graphs without the assumptionon the slack bus is not known. In this paper, we show that2-vertex-connected1 Series-Parallel graphs can be reduced toa single line without the assumption.

C. Notations

We start with some mathematical notations. We useN,Z,R,C to denote the set of all natural numbers, integers,real numbers and complex numbers, respectively. We denote[n] := {1, . . . , n} for any n ∈ N. The symbol j denotes theunit imaginary number. The notations (·)T and (·)H denote thetranspose and Hermitian transpose of a matrix, respectively.For a complex number z, |z| denotes its magnitude and fora set X , the symbol |X| denotes its cardinality.

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in the underlying graph. The underlying graph is assumedto be a simple and connected graph. The set of vertices Vand the set of edges E correspond to the set of buses andthe set of lines of the power network. The series elementof the equivalent Π-model of each line {k, `} is modeled byadmittance Yk` = Gk` − jBk`, where Gk`, Bk` ≥ 0.

We denote v ∈ Cn as the vector of complex bus voltages.The complex voltage at bus k can be written in the polar formusing its magnitude and phase angle vk = |vk|ejΘk for all k ∈[n], where |vk|∈ R and Θk ∈ R denote the voltage magnitudeand phase angle, respectively. We denote Θk` := Θk − Θ` ∈[−π, π) as the phase difference modulus by 2π for all {k, `} ∈E. In the rest of the paper, we use the corresponding valuesin [−π, π) for phase differences.D. Paper organization

The remainder of this paper is organized as follows. Sec-tion II gives the necessary background knowledge about the P -Θ power flow problem and the existing uniqueness theory forthe P -Θ problem. The notions of strong uniqueness and weakuniqueness are also introduced. In Section III, we proposethe general analysis framework of the uniqueness theory thatonly depends on the monotone regime and the topologicalstructure. We show that necessary and sufficient conditionscan be fully characterized by a feasibility problem, which hasfewer variables than the P -Θ problem. Sufficient conditionsfor uniqueness are derived and it is shown that the uniquenessconditions in [1] follow as a natural corollary. Then, weconsider three special cases in Section IV by assuming specifictopological structures for the underlying graph or a specificmonotone regime. In these special cases, the necessary andsufficient conditions are simplified and the intricate sinusoidalfunctions are avoided in the verification of those conditions.Furthermore, the sufficient conditions proposed in SectionIII are proved to be tight when no information beyond themonotone regime and the topological structure is available.Finally, a reduction method and search-based algorithms forcomputing the maximal girth and maximal eye are given inSection V.

II. PRELIMINARIESA. P -Θ problem formulation

As mentioned in the introduction, we focus our attention tothe P -Θ problem, which describes the relationship betweenthe voltage phasor angles and the real power injections. Weassume that the slack bus and the reference bus are bus 1,and that all other buses except the slack bus are PV buses.Recall that the following injection operator describes the P -Θproblem, where the shunt elements are assumed to be purelyreactive.

Definition 1. Define P̂k : {0} × Rn−1 → R as the map fromthe vector of phasor angles to the real power injection at busk:

P̂k(Θ) := 2γk` will not enlarge the set of neighboringphases compared to setting ωk` = 2γk`.

It is desirable to analyze the uniqueness of the solution inthe neighborhood N (G,Θ,W). In [1], the authors consideredthe set of allowable angles, which is defined as

{Θ̃ ∈ Rn|Θ̃1 = 0, Θ̃k` ∈ [−ωk`/2, ωk`/2],∀{k, `} ∈ E}.

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Note that the set of allowable angles is a special case of theset of allowable perturbations, since any two phase vectors inthe set of allowable angles are in the corresponding sets ofneighboring phases of each other. In this paper, we use theset of allowable perturbations but the sufficient conditions wederive can be naturally applied to using the set of allowablephases.

C. Notions of weak and strong uniqueness

Informally, we say that the P -Θ problem (1) has a uniquesolution Θ under the allowable perturbation set W , if thereexists at most one solution in the set N (G,Θ,W). We givetwo different definitions of uniqueness. Firstly, we introducethe uniqueness in the weak sense.

Definition 4. We say that a solution Θ to the P -Θ problem (1)is weakly unique with the given set of allowable perturbationsW , if for any solution Θ̃ ∈ N (G,Θ,W), there exists a line{k, `} ∈ E such that Θk` = Θ̃k`.

In other words, two solutions are different according to Def-inition 4 if and only if they have different phase differences forevery line. Next, we extend the definition of weak uniquenessto a stronger sense that is also more useful and usual.

Definition 5. We say that a solution Θ to the P -Θ problem(1) is strongly unique with the given set of allowable per-turbations W , if for any solution Θ̃ ∈ N (G,Θ,W) and any{k, `} ∈ E, we have Θk` = Θ̃k`.

In other words, two solutions are different according toDefinition 5 if and only if the phase differences are differenton at least one line. We mention that two different solutionsto problem (1) according to Definition 5 correspond to twodifferent solutions to problem (1) according to Definition 4for a “power sub-network”. To understand this, suppose thatΘ1 and Θ2 are two different solutions according to Definition5 and let ∆k` := Θ1k` − Θ2k` be the difference between thesolutions on each line {k, `} ∈ E. If ∆k` = 0 for someline {k, `} ∈ E, then we can create another power systemby removing the line {k, `} such that Θ1 and Θ2 become thesolutions to the new power system. This can be achieved byadding a generator or a load at bus k that matches the powerflow from ` to k and adding a generator or a load at bus ` thatmatches the power flow from k to `. Note that the originalsolutions Θ1 and Θ2 still solve the power flow equations forthe sub-network with some generators or loads added and theline removed. We can repeat this process until the difference∆k` becomes nonzero for all remaining lines {k, `}. If somebus becomes a singleton, we can omit it since it does notaffect the uniqueness of power flow solutions. If the slackbus becomes a singleton, we can choose another arbitrary busas the new slack bus, since the real power injection is thesame for both solutions at PV buses. Now, the two solutionsare different according to Definition 4 on the “power sub-network” of the original power network. Hence, the stronguniqueness can be implied if we ensure the weak uniquenesson all “possible sub-networks”.

III. UNIQUENESS THEORY FOR GENERAL GRAPHSIn this section, we derive necessary and sufficient condi-

tions on the set of allowable perturbations W such that thesolution to problem (1) becomes strongly or weakly unique.In particular, we aim to analyze the impact of the powersystem topology and the size of the monotone regime on theuniqueness property. Namely, given the topological structureand the monotone regime, we aim to find conditions onW such that the uniqueness of solutions holds. To achievethis, we need to derive conditions under which all powernetworks with the same topological structure and monotoneregime have unique solutions. To formalize the problem, wefix the underlying graph (V,E) and the angles specifying themonotone regime Γ := {γk` ∈ (0, π/2] | {k, `} ∈ E}. Wedefine the set of possible admittances with the same monotoneregime as

S(γ) := {(C cos(γ), C sin(γ)) | C > 0} ∀γ ∈ [0, π/2].

The set of complex admittance matrices with the same mono-tone regime is defined as

Y(V,E,Γ) := {Y is an admittance matrix |Yk` = Gk` − jBk`, (Gk`, Bk`) ∈ S(γk`), {k, `} ∈ E}.

Then, we define the set of power networks with the sametopological structure and same monotone regime as

G(V,E,Γ) := {G = (V,E, Y ) | Y ∈ Y(V,E,Γ)},

or simply G if there is no confusion about V, E and Γ. Hence,the problem under study in this paper can be stated as follows:• What are the necessary conditions and sufficient con-

ditions on the allowable perturbations W such that thesolution to problem (1) is unique within the set ofallowable perturbations for any power network G ∈ G?

The necessary conditions and the sufficient conditions providetwo sides on the uniqueness theory. The sufficient conditionsgive a guarantee for the uniqueness of solutions for anysingle power network with the given topological structure andmonotone regime, while the necessary conditions bound theoptimal conditions we can derive only using the knowledge oftopological structure and monotone regime. We first give anequivalent characterization of strong and weak uniqueness.

Lemma 1. (Necessary and Sufficient Conditions for Unique-ness) Given the set of power networks G(V,E,Γ) and the setof allowable perturbations W , the following two statementsare equivalent:

1) For any power network G ∈ G(V,E,Γ) and any powerinjection P ∈ R|V|−1 such that problem (1) is feasible inthe monotone regime, the solution to problem (1) in themonotone regime is strongly unique in N (G,Θ,W).

2) For any power network G ∈ G(V,E,Γ) and any twophase angle vectors Θ1,Θ2 in the monotone regime withthe property Θ2 ∈ N (G,Θ1,W), there exists a vectory ∈ R|V| such that y1 = 0 and

sin(γk` + Θ1k`/2 + Θ

2k`/2) · yk (2)≥ sin(γk` −Θ1k`/2−Θ2k`/2) · y`∀{k, `} ∈ E s.t. Θ1k` −Θ2k` > 0,

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where at least one of the inequalities above is strict.The equivalence between statements 1 and 2 still holds trueeven after replacing strong uniqueness with weak uniquenessin statement 1, provided that the phase angle vector Θ2 instatement 2 is required to satisfy Θ1k` 6= Θ2k` for all {k, `} ∈ E.

Intuitively, the above lemma studies the uniqueness ofsolutions through its dual form. The dual form is preferredsince the dual problem has fewer variables and its solution iseasier to construct. We then derive several sufficient conditionsusing Lemma 1. We first show that we only need to verifystatement 2 in Lemma 1 for two phase angle vectors Θ1 andΘ2 that induce a (weakly) feasible orientation, which we willdefine below. We define the orientation induced by two phaseangle vectors.

Definition 6. Suppose that Θ1 and Θ2 are two phase anglevectors of the graph. Then, we define the induced orientationof ∆ := Θ1 − Θ2 as Ak` := sign(∆k`) for all {k, `} ∈ E,where the sign function sign(·) is defined as

sign(x) :=

+1 if x > 0

0 if x = 0−1 if x < 0

.

In the definition of induced orientations, we assign one ofthe three directions +1,−1, 0 to each edge. The first twodirections are “normal” directions for directed graphs. An edgewith direction +1 or −1 is called a normal edge. Edges withdirection 0 are viewed as an undirected edge and reachablein both directions. In addition, edges with direction 0 are notconsidered when computing the in-degree and the out-degree.We only need to consider orientations induced by two differentphase angle vectors Θ1,Θ2 such that P̂ (Θ1) = P̂ (Θ2).However, a precise characterization of those orientations isdifficult and we consider a larger set that contains thoseorientations.

Definition 7. An orientation assigned to an undirected graphis called a feasible orientation if all edges are normal andeach vertex except vertex 1 has nonzero in-degree and out-degree.

According to the analysis in [1], the induced orientationof two solutions Θ1 and Θ2 in the monotone regime thatare different according to Definition 4 must be a feasibleorientation. Then, we give the definition of weakly feasibleorientations as the counterpart for strong uniqueness.

Definition 8. An orientation assigned to an undirected graphis called a weakly feasible orientation if two properties aresatisfied: (i) there exists at least one normal edge, and (ii) thein-degree and the out-degree of any vertex except vertex 1 areboth zero or both nonzero.

Edges with direction 0 are lines with the same angulardifference in two phase angle vectors. By the same discussionas in Section II, we can view a weakly feasible orientationas a feasible orientation for the sub-graph that only hasnormal edges. The next lemma shows that we only need toconsider weakly feasible orientations or feasible orientationswhen checking the conditions in statement 2 of Lemma 1.

Lemma 2. If two different phase angle vectors Θ1−Θ2 in themonotone regime satisfy Θ2 ∈ N (G,Θ1,W) and the inducedorientation of Θ1−Θ2 is not weakly feasible, then there existsa vector y ∈ R|V| such that statement 2 of Lemma 1 holds.The result holds true for the weak uniqueness property aswell, provided that the induced orientation of Θ1,Θ2 is not afeasible orientation.

Combining Lemmas 1 and 2, we obtain sufficient conditionsfor strong uniqueness and weak uniqueness.

Theorem 3. (Sufficient Conditions for Uniqueness) Given theset of allowable perturbations W , suppose that for any twodifferent phase angle vectors Θ1 and Θ2 in the monotoneregime satisfying Θ2 ∈ N (G,Θ1,W), the induced orientationof Θ1 − Θ2 is not a weakly feasible orientation. Then, thesolution to problem (1) is strongly unique for all powernetworks in G. The result holds true for the weak uniquenessas well, provided that the induced orientation of Θ1 − Θ2 isnot a feasible orientation.

The sufficient condition given above is a generalization ofTheorem 4 in [1], which ensures the weak uniqueness ofsolutions in the set of allowable phases. Using Theorem 3,we can derive a corollary similar to Theorem 4 in [1].

Corollary 4. Consider an arbitrary set of allowable perturba-tions W . The solution to problem (1) in the monotone regimeis strongly unique for any power network G ∈ G if for anyweakly feasible orientation of the underlying graph (V,E),there exists a directed cycle (k1, . . . , kt) containing at leastone normal edge such that the allowable perturbations satisfythe inequality ∑

{ki,ki+1} is normal

ωkiki+1 < 2π,

where kt+1 := k1. The same result holds true for the weakuniqueness if we substitute weakly feasible orientations withfeasible orientations.

Now, we consider a special case where all constants ωk` inthe set of allowable perturbations are equal, i.e., there exists aconstant ω ≥ 0 such that the set of allowable perturbation is

Wω := {ωk` = ω,∀{k, `} ∈ E}.

The problem we consider in this case is:• What is the sufficient condition on ω such that the

solution to problem (1) is unique with the allowableperturbation set Wω?

We derive an upper bound on the constant ω to guarantee theuniqueness. We first define the maximal eye and the maximalgirth of an undirected graph.

Definition 9. Consider an undirected graph (V,E). For anyweakly feasible orientation assigned to the graph (V,E), wedefine the minimal length of directed cycles that contain atleast one normal edge as the size of eye of this orientation,where edges with direction 0 are considered as bi-directionaledges. We define the maximal eye of the graph (V,E) as themaximum of the size of eye over all possible weakly feasible

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orientations. We denote the maximal eyes of the graph (V,E),a power network G and a group of power networks G ase(V,E), e(G) and e(G), respectively.

Remark 1. There always exists a directed cycle containingnormal edges when the underlying graph is under a weaklyfeasible orientation. To understand this, we first choose anarbitrary normal edge (k1, k2) ∈ E. Since the vertex k2 hasnonzero in-degree, it also has nonzero out-degree. Hence, thereexists another vertex k3 such that (k2, k3) ∈ E. Continuingthis procedure will result in the existence of a vertex kt suchthat vt = ks for some s < t. This generates a directed cycle(ks, ks+1, . . . , kt−1) containing only normal edges. Hence, thesize of eye is well-defined.

The counterpart of the maximal eye, known as the maximalgirth, is defined in [1] and we restate the definition below.

Definition 10. Consider an undirected graph (V,E). For anyfeasible orientation assigned to the underlying graph (V,E),we define the minimal size of directed cycles as the girthof this feasible orientation. We define the maximal girthof the graph (V,E) as the maximum of the girth over allfeasible orientations. We denote the maximal girths of thegraph (V,E), a power network G and a group of powernetworks G as g(V,E), g(G) and g(G), respectively.

Remark 2. Similar to the discussion in Remark 1, there existsat least one directed cycle when the graph is under a feasibleorientation. The maximal eye can be equivalently defined asthe maximum of the maximal girth over all sub-graphs thatdo not have degree-1 vertices.

We provide an upper bound for ω using the maximal eyeand the maximal girth, which follows from Corollary 4.

Corollary 5. If the inequality

ωk` <2π

e(G)∀{k, `} ∈ E, (3)

is satisfied, then the solution to problem (1) in the monotoneregime is strongly unique for any power network G ∈ G. Thesame result holds true for weak uniqueness, provided that e(G)in (3) is substituted by g(G).

In Section V, we design search-based algorithms to calculatethe maximal eye and the maximal girth. However, computingthe maximal eye or the maximal girth is challenging for graphswith more than 100 nodes. Hence, we seek upper bounds andlower bounds for the maximal eye and the maximal girth.In this paper, we obtain a simple upper bound for both themaximal girth and the maximal eye. We define κ(G) and κ(G)as the sizes of the longest chordless cycles of the underlyinggraph of the power network G and any power network in thepower network class G, respectively. The upper bound on themaximal girth and eye will be provided below.

Theorem 6. For any power network G, it holds that

g(G) ≤ e(G) ≤ κ(G) (4)

and that g(G) ≤ e(G) ≤ κ(G).

We note that although computing the longest chordless cycleisNP-complete [23], the computation of the longest chordlesscycle is faster than the computation of the maximal eye andthe maximal girth in practice.

IV. UNIQUENESS THEORY FOR THREE SPECIAL CASES

In this section, we consider three special cases. For eachcase, the power network either has a special topologicalstructure or a special monotone regime. In the first two cases,the underlying graph of the power network is a single cycleor a 2-vertex-connected Series-Parallel (SP) graph. When theunderlying graph is a single cycle, the sufficient conditionsin Corollary 4 are also necessary. If the underlying graph isa 2-vertex-connected SP graph, we prove that the sufficientconditions for the weak uniqueness in Corollary 5 also ensurethe strong uniqueness. In the last case, the power networkis assumed to be lossless. In this case, the monotone regimeof each line reaches the maximum possible size [−π/2, π/2].Sinusoidal functions can then be avoided in statement 2 ofLemma 1, and therefore the verification of conditions is easier.

A. Single cycles

We first consider the case when the underlying graph (V,E)is a single cycle. We first show that the weak uniqueness isequivalent to the strong uniqueness in this case.

Lemma 7. Suppose that the underlying graph is a single cyclewith the edges (1, 2), (2, 3), . . . , (n, 1). Then, given the set ofallowable perturbations W , the solution to problem (1) in themonotone regime is weakly unique if and only if it is stronglyunique.

Next, we prove that the sufficient conditions derived inCorollary 4 are also necessary for a single cycle with non-trivial monotone regime.

Theorem 8. Suppose that the underlying graph is a singlecycle with the edges (1, 2), (2, 3), . . . , (n, 1), and that the setof allowable perturbations satisfies 0 < ωi,i+1 ≤ γi,i+1 forall i ∈ [n], where γn,n+1 := γn,1 and ωn,n+1 := ωn,1. Thesolution to problem (1) in the monotone regime is stronglyunique for any power network G ∈ G(V,E,Γ) and any powerinjection P ∈ Rn−1 that makes problem (1) feasible if andonly if the set of allowable perturbations W satisfies

n∑i=1

ωi,i+1 < 2π,

where ωn,n+1 := ωn,1.

In contrast to requiring ωi,i+1 > 0 in the above theorem, thecondition that ωi,i+1 = 0 for some i is sufficient but not nec-essary for the uniqueness of solutions. Under this condition,two solutions Θ1 and Θ2 in the monotone regime such thatΘ2 ∈ N (G,Θ1,W) must satisfy Θ1i,i+1 = Θ2i,i+1. Hence,any solution is strongly unique with this set of allowableperturbations. However, by Theorem 8, this condition is notnecessary for the uniqueness of solutions.

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B. Series-Parallel graphs

In this subsection, we consider another special class ofgraphs, namely, the 2-vertex-connected SP graphs. The objec-tive is to find an upper bound on the constant ω to guaranteethat the solution to problem (1) is unique. Corollary 5 showsthat the solution is strongly unique if ω < 2π/e(G) and isweakly unique if ω < 2π/g(G). However, for a 2-vertex-connected SP graph, we can prove a stronger theorem. Wefirst prove that the maximal eye is equal to the maximal girthfor a 2-vertex-connected SP graph. The main tool is the eardecomposition of an undirected graph [27].

Definition 11. An ear of an undirected graph (V,E) is asimple path or a single cycle. An ear decomposition of anundirected graph (V,E), denoted as D := (L0, . . . , Lr−1),is a partition of E into an ordered sequence of ears suchthat one or two endpoints of each ear Lk are contained inan earlier ear, i.e., an ear L` with ` < k, and the internalvertices of each ear do not belong to any earlier ear. We callD a proper ear decomposition if each ear Lk is a simplepath for all k = 1, . . . , r − 1. A tree ear decomposition isa proper ear decomposition in which the first ear is a singleedge and for each subsequent ear Lk, there is a single earL` with ` < k, such that both endpoints of Lk lie on L`. Anested ear decomposition is a tree ear decomposition suchthat, within each ear L`, the set of pairs of endpoints of otherears Lk that lie within L` forms a set of nested intervals.

The following theorem in [28] provides an equivalent char-acterization of 2-vertex-connected SP graphs through the eardecomposition.

Theorem 9. A 2-vertex-connected graph is series-parallel ifand only if it has a nested ear decomposition.

With the help of the nested ear decomposition, we will provethat the maximal girth is equal to the maximal eye for 2-vertex-connected SP graphs. The intuition behind the proof is that wefirst choose two vertices as the “source” and the “sink” forthe power flow network, and then assign a normal direction toeach edge with direction 0 according to the directed path thatcontains this edge and goes from the “source” to the “sink”.This makes the first inequality in (4) holds as equality.

Lemma 10. Suppose that (V,E) is a 2-vertex-connected SPgraph. Then, the following equality holds true:

g(V,E) = e(V,E).

Therefore, combining the above lemma with Corollary 5, weobtain a stronger sufficient condition for 2-vertex-connectedSP graphs. This result implies that the sufficient conditionsfor the weak uniqueness in Corollary 5 also guarantee thestrong uniqueness.

Theorem 11. Suppose that the underlying graph (V,E) is a2-vertex-connected SP graph. The solution to problem (1) isstrongly unique for any power network G ∈ G in the monotoneregime if

ω <2π

g(G).

C. Lossless networks

Finally, we consider the case when the power networkis lossless, namely, when γk` = π/2 for all {k, `} ∈ E.In this case, we prove that the strong uniqueness holds ifand only if there does not exist another solution in the setof neighboring phases such that the induced orientation hasstrictly more strongly connected components than weaklyconnected components. This results makes it possible to avoidnonlinear sinusoidal functions in statement 2 of Lemma 1,and therefore the uniqueness of solutions becomes easier toverify. We first define the sub-graph induced by two phaseangle vectors.

Definition 12. Suppose that Θ1 and Θ2 are two differentphase angle vectors, and that the orientation A is the inducedorientation of Θ1 − Θ2. Then, the induced sub-graph ofΘ1 −Θ2 is constructed as a directed sub-graph of (V,E, A)by first deleting all edges with direction 0 and then deletingall degree-1 vertices.

In what follows, we establish a necessary and sufficientcondition for the uniqueness of the solution that does notcontain sinusoidal functions.

Theorem 12. Consider a that the set of allowable perturba-tions W . If the monotone regime satisfies γk` = π/2 for all{k, `} ∈ E, then the following two statements are equivalent:

1) For any power network G ∈ G(V,E,Γ) and any powerinjection P ∈ R|V|−1 such that problem (1) is feasible,the solution to problem (1) in the monotone regime isstrongly unique in N (G,Θ,W).

2) For any power network G ∈ G(V,E,Γ) and any twophase angle vectors Θ1 − Θ2 in the monotone regimewith the property Θ2 ∈ N (G,Θ1,W), the induced sub-graph of Θ1 − Θ2 has strictly more strongly connectedcomponents than weakly connected components.

The equivalence between statements 1 and 2 still holds trueeven after replacing strong uniqueness with weak uniquenessin statement 1, provided that the phase angle vectors Θ2 instatement 2 is required to satisfy Θ1k` 6= Θ2k` for all {k, `} ∈ E.

The result of the above theorem is stronger than the suffi-cient conditions in Theorem 3. This is because any (weakly)infeasible orientation has strictly more strongly connectedcomponents than weakly connected components. Hence, thesufficient conditions in Theorem 3 ensure that all inducedorientations are (weakly) infeasible. Then, statement 2 of thistheorem holds true and the solution becomes strongly (weakly)unique.

V. ALGORITHMS AND NUMERICAL RESULTS

In the preceding sections, we have shown that the maxi-mal eye and the maximal girth play important roles in theuniqueness theory. However, computing the maximal eye ormaximal girth is cumbersome for large graphs. Hence, wedevelop a successive reduction method to design a reducedgraph, and then prove the relationship between the maximaleye or the maximal girth of the original graph and those of thereduced graph. Next, we propose search-based algorithms for

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computing the maximal eye and the maximal girth. Finally,we test the performance of those algorithms on real-worldproblems.

A. Successive Series-Parallel Reduction method

In this subsection, we propose a successive reductionmethod, named as the Successive Series-Parallel Reduction(SSPR) method, that can reduce the size of the underlyinggraph. for computing the maximal eye and maximal girth. TheSSPR method is different from the Series-Parallel Reduction(SPR) method introduced in [1] in two aspects. First, thepurpose of the SSPR method is to accelerate the computationof the maximal eye and the maximal girth, while the focusof SPR method is to facilitate the verification of uniquenessconditions. Second, we prove that all 2-vertex-connected SPgraphs can be reduced to a single edge (K2) without makingthe assumption in [1] that the slack bus is the last bus to bereduced.

Before introducing the SSPR method, we extend the defi-nition of the maximal eye and the maximal girth to weightedgraphs with “multiple slack buses”. This generalized class ofgraphs appear during the reduction process. By defining thelength of a cycle as the sum of the weights of the edgeson the cycle, the maximal eye and the maximal girth canbe generalized to weighted graphs. Next, we define (weakly)feasible orientations for graphs with “multiple slack buses”,namely, the slack nodes.

Definition 13. For a weighted undirected graph (V,E,W ), asubset of vertices Vs ⊆ V is called the set of slack nodes.An orientation A assigned to the graph is called a weaklyfeasible orientation if each edge has one of the directions{+1,−1, 0} and each vertex not in Vs either has nonzero in-degree and nonzero out-degree, or has zero in-degree and zeroout-degree. An orientation A assigned to the graph is calleda feasible orientation if each edge has one of the directions{+1,−1} and each vertex not in Vs has nonzero in-degreeand nonzero out-degree.

Now, we can define the maximal eye for graphs with slacknodes by taking the maximum of the size of eye over weaklyfeasible orientations. The maximal girth can be defined ina similar way. For power networks, the only slack node isthe slack bus of the power network. Hence, the extendeddefinitions of the maximal eye and the maximal girth areconsistent with their original definitions. The SSPR methodis based on three types of operations:• Type I Operation. Replacement of a set of parallel edges

with a single edge that connects their common endpoints.The weight of the new single edge is the minimum overthe weights of the deleted parallel edges.

• Type II Operation. Replacement of the two edges in-cident to a degree-2 vertex with a single edge, if thevertex has exactly two neighboring vertices and is nota slack node. The weight of the new edge is the sum ofthe weights of the two deleted edges.

• Type III Operation. Deletion of a vertex that has only asingle neighboring vertex. If the deleted vertex is a slacknode, or if the deleted vertex has degree at least 2 for the

Algorithm 1 Successive Series-Parallel Reduction (SSPR)methodInput: Undirected unweighted graph (V,E), slack bus kOutput: Reduced undirected weighted graph (VR,ER,WR),

two constants α1, α2 defined in Theorems 14 and 16, set ofslack nodes VsSet the initial weight for each edge to be 1.Set the initial set of slack nodes as Vs ← {k}.while at least one operation is implementable do

if Type I Operations are implementable thenImplement Type I Operation.Update values α1, α2 according to their definitions

in Theorems 14 and 16.continue

end ifif Type II Operations are implementable then

Implement Type II Operation.continue

end ifif Type III Operations are implementable then

Implement Type III Operation.Update values α1, α2 according to their definitions

in Theorems 14 and 16.Update the set of slack nodes Vs.continue

end ifend whileReturn reduced graph (VR,ER,WR), set of slack nodes Vsand values α1, α2.

problem of computing the maximal girth, then we defineits neighboring vertex as a slack node.

The update scheme of weights and slack nodes is designedto control the change of the maximal eye or the maximalgirth. The SSPR method successively reduces the size of thegraph by applying Type I-III Operations; the pseudo-code ofthe SSPR method is given in Algorithm 1. We note that afterthe reduction process, there is no parallel edge or pendant(degree-1) vertex in the reduced graph. Ignoring the weightsof the edges and the set of the slack nodes, the operationsin the SSPR method can cover the operations in the classicalseries-parallel reduction [26], which are defined as

• Type I’ Operation. Replacement of parallel edges witha single edge that connects their common endpoints.

• Type II’ Operation. Replacement of the two edgesincident to a degree-2 vertex with a single edge.

• Type III’ Operation. Deletion of a pendant vertex.

Hence, the SSPR method can be viewed as a generalizationof the classical series-parallel reduction. Now, we study themaximal eye of the graph before and after the reduction.We first consider the change of the maximal eye after eachoperation.

Lemma 13. Given a weighted undirected graph (V,E,W ),let e denote its maximal eye. Assume that one of Type I-IIIOperations is implemented on the graph. By denoting the new

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graph and its maximal eye as (Ṽ, Ẽ, W̃ ) and ẽ, the followingstatements hold:• If Type I Operation is implemented, then

ẽ ≤ e ≤ max{ẽ,Wmax +Wmin},

where Wmax and Wmin are the maximal and minimalweights of the deleted parallel edges, respectively.

• If Type II Operation is implemented, then e = ẽ.• If Type III Operation is implemented and the deleted

vertex has degree 1, then e = ẽ.• If Type III Operation is implemented and the deleted

vertex has degree larger than 1, then

e = max{ẽ,Wmax +Wmin},

where Wmax and Wmin are the maximal and minimalweights of the deleted parallel edges, respectively.

Using the above lemma, we have the following theorem.

Theorem 14. Given a power network with the underlyinggraph (V,E), let e denote the maximal eye of the graph.Denote the graph after reduction and its maximal eye as(VR,ER,WR) and eR, respectively. Then, we have

max{eR, α2} ≤ e ≤ max{eR, α1, α2},

where α1 and α2 are the maximum of Wmax+Wmin over TypeI and Type III Operations, respectively. Here, Wmax,Wminare defined in Lemma 13. If Type I or Type III Operations isnot implemented, then we set α1 or α2 to 0.

Similarly, we can prove the relation between the maximalgirth of the original graph and that of the reduced graph. Wefirst show the change of the maximal girth after each operation.

Lemma 15. Given a weighted undirected graph (V,E,W ),let g denote its maximal girth. Assume that one of Type I-IIIOperations is implemented on the graph. By denoting the newgraph and its maximal girth of new graph as (Ṽ, Ẽ, W̃ ) andg̃, the following statements hold:• If Type I Operation is implemented, then

g̃ ≤ g ≤ max{g̃,Wmax +Wmin},

where Wmax and Wmin are the maximal and minimalweights of the deleted parallel edges, respectively.

• If Type II Operation is implemented, then g = g̃.• If Type III Operation is implemented and the deleted

vertex has degree 1, then g = g̃.• If Type III Operation is implemented, the deleted vertex

is a slack node and has degree larger than 1, then

g̃ ≤ g ≤ max{g̃,Wmax +Wmin},

where Wmax and Wmin are the maximal and minimalweights of the deleted parallel edges, respectively.

• If Type III Operation is implemented, the deleted vertexis not a slack node and has degree larger than 1, then

g = min{g̃,Wmax +Wmin},

where Wmax and Wmin are the maximal and minimalweights of the deleted parallel edges, respectively.

By the above lemma, the relationship between the maximalgirth of the original graph and that of the reduced graph willbe discovered below.

Theorem 16. Given a power network with the underlyinggraph (V,E), let g denote its the maximal girth. By de-noting the graph after reduction and its maximal girth as(VR,ER,WR) and gR, we have

min{gR, α2} ≤ g ≤ min{max{gR, α1}, α2},

where α1 is the maximum of Wmax + Wmin over Type IOperations and the second case of Type III Operations, and α2is the minimum of Wmax +Wmin over the third case of TypeIII Operations. Here, Wmax,Wmin are defined in Lemma 13.If operations for computing α1 or α2 are not implemented,then we set α1 to 0 or α2 to +∞.

Based on the numerical results in Tables I and II for largepower networks, the values of α1 and α2 in Theorems 14 and16 are usually smaller than eR and gR. Hence, we have theapproximation

e ≈ eR, g ≈ α2. (5)

The above relations imply that for large power networks,computing the maximal eye is equivalent to computing themaximal eye of a reduced graph, while the maximal girth isalready computed during the reduction process. Finally, weprove that 2-vertex-connected SP graphs can be reduced to asingle edge by the SSPR method.

Theorem 17. If the underlying graph (V,E) of a powernetwork is a 2-vertex-connected SP graph, then the SSPRmethod reduces the underlying graph to a single edge.

For an undirected graph without slack nodes, the classicalseries-parallel reduction (Type I’-III’ Operations) can reducethe graph to a single edge if and only if the graph is aGeneralized Series-Parallel (GSP) graph [26]. We note that 2-vertex-connected SP graphs are a special class of GSP graphsand it is unclear whether the reduction guarantee for theSSPRmethod can be extended to any GSP graphs in the presenceof slack nodes.

B. Algorithms for computing maximal eye and maximal girth

In this subsection, we propose search-based algorithmsfor computing the maximal eye and the maximal girth. Ourapproach is based on the Depth-First Search (DFS) methodand utilized the pruning technique to accelerate the computingprocess. We first describe a common sub-procedure that willbe used in both algorithms. The sub-procedure computes theminimal directed chordless cycle containing a given edge.Given a truncation length T ≥ 1, the sub-procedure returns thetruncation length if there does not exist a directed chordlesscycle that contains the given edge and has length at mostT . The sub-procedure is also based on the DFS methodwith pruning and borrows the idea of blocking from [29] toaccelerate the searching process. The pseudo-code of the sub-procedure is listed in Algorithm 2.

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The search space of the sub-procedure is the set of directedchordless paths with length at most T . When the currentdirected chordless path is a directed chordless cycle, the lengthof the cycle is recorded and the minimal length of knowndirected chordless cycles is updated. By searching over allchordless paths, we find the length of the minimal directedchordless cycle. The DFS method is initialized with the givenedge, denoted as (k, `), and extends the directed chordless pathby adding a neighbouring vertex of the end point other thank to the path. The pruning technique becomes effective anddelete the end point other than k from the path if one of thefollowing cases occurs:• The length of the directed chordless path is larger than T

or the known minimal length of directed chordless cycles;• All neighbouring vertices have been searched or will

introduce a chord if added to the path.Using the idea of blocking, one can efficiently check whetheradding a vertex to the path will introduce a chord. Thisapproach is based on the following observation: if the path(k1, . . . , kt) is chordless, then any vertex ks can only be inthe neighborhood of ks−1, ks+1. We construct an array and,for each vertex, we record the number of vertices on the paththat are in the neighborhood of the vertex. The array is updatedwhenever the path is updated. If there are at least two verticeson the path in the neighbourhood of a vertex not on the path,then adding the vertex to the path will introduce a chord.Hence, the cost of checking this condition for each potentialvertex not on the path is a single evaluation of an array.

Next, we propose the algorithms for computing the maximaleye and the maximal girth. Since the algorithm of maximalgirth is similar to the algorithm for maximal eye, we onlypresent the algorithm for computing the maximal eye and offerthe other one in the appendix. The algorithm is also based onthe DFS method with pruning, and the pseudo-code is providedin Algorithm 3. We first order all edges and gradually assignone of the directions {0,−1,+1} to each edge followingthe ordering of the edges. The search space consists of theorientations for the first several edges (intermediate states)and the orientations for the entire graph (final states). Onecan verify that all intermediate states and final states form atrinomial2 tree, since each orientation for the first k < |E|edges leads to three different orientations for the first k + 1edges. Then, the algorithm for computing the maximal eyesearches in the same way as the classical DFS method ona directed tree. For each node, we consider the sub-graphconsisting of those edges that have been assigned a direction.We compute the length of the minimal directed chordless cyclein the sub-graph, which contains the last edge in the sub-graph,using the sub-procedure (Algorithm 2). The truncation lengthcan be decided as follows. Since a DFS method is implementedon a trinomial tree, there exists a directed path from the rootnode of the trinomial tree to the current node. The truncationlength can be chosen as the minimal length computed on thepreceding nodes of the path. When the search reaches a leafnode, we obtain an orientation for the entire graph, and the

2A directed tree is called a trinomial tree if there is a root node and eachnon-leaf node has exactly three descendant nodes.

Algorithm 2 Truncated Minimal Chordless CycleInput: Directed weighted graph (V,E,W ), selected edge

(k, `), truncation length TOutput: Length of minimal chordless cycle c

Construct the neighbourhood of each vertex N : V 7→ 2V.Initialize blocked array block[i]← 0 for all vertices i ∈ V.Set the length of minimal cycle recorded c← T .Set current length Lcur ←Wk`.Set the path P ← [k, `].Set block[k]← 1, block[`]← 1.if Lcur ≥ T then . Already longer than truncation length

return cend ifwhile the length of P is at least 2 do

Get the endpoint i← P [−1].Increase block for vertices in N [j] by 1.Get the minimal vertex j ∈ N [i] such that block[j] ≤ 1

and Lcur +WP [−1]j < v.if no such vertex j exists then

. Recursion: no next unblocked vertexFind the maximal index h such that P [h] /∈ {k, `, i}

and P [h+ 1] is not the maximal vertex in N [P [h]].if no such h exists then . Search finished

breakelse

Remove P [h+ 1], . . . , P [−1] from path P .Decrease block of N [P [h]], . . . , N [P [−1]] by 1.Add the next smallest vertex in N [P [h]] to P .Update Lcur to be the length of path P .continue

end ifelse . Add a new vertex

Add vertex j to P and update Lcur.if k ∈ N [j] then . find a cycle

Calculate length ccur ← Lcur +Wjk.if ccur > 0 then

Update c← min{c, ccur}.end ifRecursion similarly as above.

elsecontinue

end ifend if

end whilereturn c

size of the eye becomes the minimal length on the path tothe root node. By searching over all leaf nodes, we find themaximal eye. Similarly, one can use the pruning technique toreduce the search space. The current node is pruned if it cannot be extended to a weakly feasible orientation for the entiregraph, or the size of the eye of the sub-graph is smaller thanthe known maximal size of the eye.

C. Numerical results

In this subsection, we verify the theoretical results of thiswork and test the performance of the proposed algorithms.

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Algorithm 3 Algorithm for Computing The Maximal EyeInput: Undirected weighted graph (V,E,W ), slack bus kOutput: Maximal eye e

Set the maximal eye e← 0.Assign an order to the set of edges E and denote edges as

{k1, `1}, . . . , {km, `m}.

Initialize the set of edges E0 ← {{k1, `1}}.Initialize the set of orientations Ak1,`1 ← −1.loop

Check the weak feasibility with current orientation.if weak feasibility fails then . Recursion

Get the maximal index j such that Akj ,`j 6= 1.if no such j exists then . Terminate the loop

breakelse

Remove {kj+1, `j+1}, . . . , {km, `m} from E0.Change orientation Akj ,`j ← Akj ,`j + 1.continue

end ifend ifCompute the size of eye ecur under E0 and A using

Algorithm 2. The truncation length is set to be the size ofeye of the precedent state.

if ecur < e then . Smaller than known size of eyeRecursion in the same way.

end ifGet the next edge {ki, `i} that is not in E0.if no such edge then . Leaf node reached

Update e← max{e, ecur}.Recursion in the same way.

elseAdd the next edge {ki, `i} that is not in E0.Assign Akj ,`j ← −1.continue

end ifend loopreturn e

First, we show that, using the SSPR method, the computationof the maximal eye can be reduced to a smaller graph, whilethe computation of the maximal girth is finished during theprocess of reduction. Then, we show that Corollary 5 gives avalid sufficient condition for strong uniqueness. We use IEEEpower networks in MATPOWER [30] to perform experiments.

We first consider the computation of the maximal eye. Theresults are listed in Table I. Here, we use ‘-’ to denote thecase when this value does not exist, and use ‘TLE’ (Time LimitExceeded) to denote the case when the algorithm does not findany leaf node in two days. The lower bounds for the maximaleye are derived by stopping the algorithm before it terminates.It can be observed that the SSPR method can largely reducethe size of the graph, and therefore can accelerate the com-puting process. Moreover, the values of α1 and α2 are smallcompared to the maximal eye of the reduced graph. Hence, theapproximation in equation (5) holds and the maximal eye of

Power Network Original Size Reduced Size α1 α2 eRCase 14 (14,20) (2,1) 6 3 0Case 30 (30,41) (8,13) 4 3 8Case 39 (39,46) (8,12) 4 5 8Case 57 (57,78) (22,39) 4 - 23

Case 118 (118,179) (44,83) 5 - 13Case 300 (300,409) (109,196) 8 4 ≥10Case 1354 (1354,1710) (263,500) 9 8 TLECase 2383 (2383,2886) (499,949) 11 5 TLE

TABLE I: Number of vertices and edges before and after the SSPRmethod for maximal eye along with values computed during thereduction process.

the original graph is equal to the maximal eye of the reducedgraph. Although the algorithm achieves acceleration comparedto the brute-force search method, we are only able to computethe maximal eye for graphs with up to 118 vertices. Note thatsince graph problems have exponential complexities, solvingthem for graphs having as low as 200 nodes is still beyondthe current computational capabilities. However, this does notundermine the usefulness of the introduced graph parameters,since it is shown in this work that those parameters accuratelydecide whether the power flow problem has a unique solution.

Next, we consider the computation of the maximal girth. Weuse the same algorithms and the results are listed in Appendix.In this case, it can be observed that α2 is equal to 3 forlarge power networks. This is because the underlying graphsof large power networks considered in the table have “pendanttriangles”. Pendant triangles are triangles that have only onevertex connected to the rest of the graph. Furthermore, theapproximation in Theorem 16 holds and the maximal girth ofthe original graph is equal to α2 = 3. Hence, the maximal girthcan be computed during the reduction process. This shows thatthe conditions for the weak uniqueness is significantly looseand requiring ωk` to be at most 2π/3 for all edges {k, `} isenough. However, for 2-vertex-connected SP graphs, we haveshown that the maximal girth is equal to the maximal eye andthe requirement for the weak uniqueness is the same as thatfor the strong uniqueness.

Finally, we validate the results in Corollary 5, i.e., showingthat there does not exist a different solution in the monotoneregime with the set of allowable perturbations beingW2π/e(G).Given a power network in MATPOWER data set, we randomlygenerate power systems with the same network topologyand monotone regime. Then, we construct several randomsolutions and use MATPOWER runpf to check if there aredifferent solutions in the neighborhood of the constructedground truth solution. For power systems with at most 118buses, the ground truth solution is the only solution foundin the neighborhood of the ground truth. Hence, we knowthe strong uniqueness holds for those power networks. Moredetails about the experimental setup are given in the appendix.

VI. CONCLUSIONIn this paper, we extend the uniqueness theory of P -Θ

power flow solutions developed in [1] for an AC power system.The notion of strong uniqueness is introduced to characterizethe uniqueness in the common sense. We propose a generalnecessary and sufficient condition for the uniqueness of the

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solution, which depends only on the monotone regime and thenetwork topology. These conditions can be greatly simplifiedin certain scenarios. When the underlying graph of the powernetwork is a single cycle, sufficient conditions in [1] are provedto be necessary. For 2-vertex-connected SP graphs, we showthat the maximal eye is equal to the maximal girth, whichmeans that the sufficient condition for the weak uniquenessalso implies the strong uniqueness. When the power networkis lossless, we derive a necessary and sufficient condition thatdoes not contain sinusoidal functions and its sufficient partis stronger than the general sufficient conditions. A reductionmethod, named the SSPR method, is proposed to reduce thesize of power network and accelerate the computation of themaximal eye and the maximal girth. The SSPR method isproved to reduce a 2-vertex-connected SP graph to a singleedge and the relation between the graphs before and afterthe reduction is analyzed. Some algorithms based on the DFSmethod with pruning are designed to compute the maximaleye and maximal girth.

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[30] R. Zimmerman, C. Murillo-Sanchez, and R. Thomas, “Matpower:Steady-state operations, planning and analysis tools for power systemsresearch and education,” IEEE Transactions on Power Systems, vol. 26,no. 1, pp. 12–19, 2011.

Haixiang Zhang received the B.S. degree in Applied Mathematicsand Computer Science from Peking University, China, in 2019. Heis currently a Ph.D. student in the Department of Mathematics fromthe University of California, Berkeley, CA.

SangWoo Park received the B.S. degree in Environmental Engineer-ing from Johns Hopkins University, Baltimore, MD, in 2016 and theM.S. degree in Industrial Engineering and Operations Research fromthe University of California, Berkeley, CA, in 2017. He is currentlya Ph.D. candidate in the Department of Industrial Engineering andOperations Research from the University of California, Berkeley, CA.

Javad Lavaei received the Ph.D. degree in control and dynamicalsystems from the California Institute of Technology, Pasadena, CA,in 2011. He is an Associate Professor in the Department of Indus-trial Engineering and Operations Research, University of California,Berkeley, Berkeley, CA. Dr. Lavaei received multiple awards, in-cluding the NSF CAREER Award, Office of Naval Research YoungInvestigator Award, and the Donald P. Eckman Award.

Ross Bladick received the B.Sc. degree in mathematics and physicsand the B.E. degree in electrical engineering from the Universityof Sydney, Sydney, and the M.S. and Ph.D. degrees in electricalengineering and computer sciences from the University of California,Berkeley, CA in 1988 and 1990, respectively. He is an Emeritus Pro-fessor with the Department of Electrical and Computer Engineering,University of Texas, Austin.

https://lavaei.ieor.berkeley.edu/Mono_PF_2019_1.pdfhttps://arxiv.org/pdf/1904.08855.pdfhttps://arxiv.org/pdf/1506.08472.pdfhttps://arxiv.org/pdf/1901.11189.pdf

13

APPENDIX

A. Proof of Lemma 1

Proof. We only prove the strong uniqueness part since theproof for weak uniqueness is similar. For a given power net-work, we define the real power flow along the line {k, `} ∈ Efrom bus k in the direction of bus ` as

p̃k`(Θ) := −Gk`|vk||v`|cos(Θk`) +Bk`|vk||v`|sin(Θk`).

By definition, it follows that

P̂k(Θ) =∑

`:{k,`}∈E

p̃k`(Θ) ∀k ∈ V.

a) Proof of sufficiency: We first show by contradictionthat statement 2 of the lemma is sufficient for statement 1.In particular, suppose that statement 2 holds, but the solutionis not strongly unique for some graph G ∈ G and some realpower injection P while problem (1) is feasible. Then, thereexist two different phase angle vectors Θ1,Θ2 such that Θ2 ∈N (G,Θ1,W) and P̂ (Θ1) = P̂ (Θ2). For each line {k, `} ∈ E,there exists a constant Ck` > 0 such that

Bk` = Ck` sin(γk`), Gk` = Ck` cos(γk`).

We calculate the change of power flow from k to ` as

p̃k`(Θ1)− p̃k`(Θ2)

= −Gk`|vk||v`|[cos(Θ1k`)− cos(Θ2k`)]+Bk`|vk||v`|[sin(Θ1k`)− sin(Θ2k`)]

= −Ck` cos(γk`)|vk||v`|[cos(Θ1k`)− cos(Θ2k`)]+ Ck` sin(γk`)|vk||v`|[sin(Θ1k`)− sin(Θ2k`)]

= (− cos(γk`)[cos(Θ1k`)− cos(Θ2k`)]+ sin(γk`)[sin(Θ

1k`)− sin(Θ2k`)]) · |vk||v`|Ck`

= 2[cos(γk`) sin(Θ1k`/2 + Θ

2k`/2)

+ sin(γk`) cos(Θ1k`/2 + Θ

2k`/2)]

· sin(∆k`/2)|vk||v`|Ck`= 2 sin(γk` + Θ

1k`/2 + Θ

2k`/2) · sign(sin(∆k`/2))· |sin(∆k`/2)||vk||v`|Ck`

:= δk` · |sin(∆k`/2)vkv`|Ck`,

where

∆k` := Θ1k` −Θ2k`, (6)

δk` := 2 sin(γk` + Θ1k`/2 + Θ

2k`/2)sign(sin(∆k`/2)).

Note that the third equality in (6) is due to the followingtriangular identities:

cos(η)− cos(ϕ) = −2 sin[(η − ϕ)/2] sin[(η + ϕ)/2],sin(η)− sin(ϕ) = 2 sin[(η − ϕ)/2] cos[(η + ϕ)/2].

Since P̂k(Θ1) = P̂k(Θ2) for all k 6= 1, we obtain

P̂k(Θ1)− P̂k(Θ2) =

∑`:{k,`}∈E

[p̃k`(Θ

1)− p̃k`(Θ2)]

=∑

`:{k,`}∈E

δk` · |sin(∆k`/2)vkv`|Ck` = 0

for all k 6= 1. Let the set E0 be the subset of edges such that∆k` 6= 0 for all {k, `} ∈ E0; we assign an order to elements inE0. Define the matrix M ∈ R|V|×|E0| and the vector g ∈ R|E0|as

Mki := δk`, M`i := δ`k, gi := |sin(∆k`/2)vkv`|Ck`,

where {k, `} is the i-th edge in the set E0. Since ∆k` 6= 0 forall {k, `} ∈ E0 and ∆k` ≤ 2γk` ≤ π, it holds that

|sin(∆k`/2)|> 0 ∀{k, `} ∈ E0.

Then, the vector g is a solution to the linear feasibility problem

find x ∈ R|E0| s.t. (Mx)2:|V | = 0, x > 0.

where (y)i:j := (yi, yi+1, . . . , yj) includes the i-th to the j-thentries of the vector y and inequality x > 0 means that xk > 0holds for all entries of the vector x. The notation x ≥ 1 isdefined in the same way. The above feasibility problem isequivalent to

find x ∈ R|E0| s.t. (Mx)2:|V | = 0, x ≥ 1.

Then, by Farka’s Lemma, the dual feasibility problem

find y ∈ R|V| s.t. MTy ≥ 0, 1TMTy > 0, y1 = 0

is infeasible. However, the conditions in the dual problem arethe same as the conditions in statement 2 of Lemma 1. Thiscontradicts the claim in statement 2 that there exists a vector ysatisfying these conditions. Thus, statement 1 must hold true.

b) Proof of necessity: Next, we again show by contradic-tion that statement 2 of the lemma is necessary for statement1. Assume that statement 1 holds true, and that there exist twodifferent phase angle vectors Θ1,Θ2 in the monotone regimesuch that Θ2 ∈ N (G,Θ1,W) while there does not exist ysatisfying the conditions in statement 2. We define E0 as theset of edges such that ∆k` 6= 0, where ∆k` := Θ1k` −Θ2k` forall {k, `} ∈ E0. We construct the matrix M ∈ R|V|×|E0| as

Mki := δk`, M`i := δ`k,

where {k, `} is the i-th edge in the set E0 and

δk` := sin(γk` + Θ1k`/2 + Θ

2k`/2)sign(sin(∆k`/2)).

By the same analysis, the conditions in statement 2 turn out tobe equivalent to the feasibility of the linear feasibility problem

find y ∈ R|V| s.t. MTy ≥ 0, 1TMTy > 0, y1 = 0.

By our assumption, the above problem is infeasible. By Farka’sLemma, there exists a solution g ∈ R|E0| to the feasibilityproblem

find x ∈ R|E0| s.t. (Mx)2:|V | = 0, x ≥ 1

and also to the feasibility problem

find x ∈ R|E0| s.t. (Mx)2:|V | = 0, x > 0.

We define the matrix C ∈ R|V|×|V| as

Ck` := |sin(∆k`/2)vkv`|−1gi ∀{k, `} ∈ E0,

where {k, `} is the i-th edge in the set E0, and

Ck` := 1 ∀{k, `} ∈ E\E0, Ck` := 0 ∀{k, `} /∈ E.

14

By the definition, it follows that Ck` > 0 for all {k, `} ∈ E.We construct a graph G with the complex admittance matrix

Yk` := Ck` cos(γk`)− jCk` sin(γk`) ∀{k, `} ∈ E.

Then, for all k 6= 1, we have

P̂k(Θ1)− P̂k(Θ2) =

∑`:{k,`}∈E

[p̃k`(Θ

1)− p̃k`(Θ2)]

=∑

`:{k,`}∈E

δk` · |sin(∆k`/2)vkv`|Ck` = (Mg)k = 0.

This implies that Θ1 and Θ2 are both solutions to problem (1)in the monotone regime when the real power injection is

P := P̂ (Θ1).

This contradicts statement 1 that the solution is strongly uniquefor any real power injection. Hence, the conditions in statement2 must be satisfied.

B. Proof of Lemma 2

Proof. We only prove the strong uniqueness part since theproof for weak uniqueness is similar. Since the inducedorientation A is not a weakly feasible orientation, there existsa vertex i 6= 1 such that it has nonzero out-degree and zeroin-degree, or it has nonzero in-degree and zero out-degree.Without loss of generality, assume that the vertex i has nonzeroout-degree and zero in-degree. We prove that the i-th unitvector y := ei satisfies the conditions in statement 1 ofLemma 1. It is straightforward that y1 = 0. We only needto show that the inequalities in (2) hold and at least one ofthem is strict. We consider any edge (k, `) such that ∆k` > 0.First, if k 6= i and ` 6= i, then both sides of the inequality (2)are zero. Next, if k 6= i and ` = i, then the condition ∆ki > 0implies that Aki = +1, which contradicts the assumption thati has zero in-degree. Finally, if k = i and ` 6= i, the goal isto prove that

sin(γi` + Θ1i`/2 + Θ

2i`/2) · yi> sin(γi` −Θ1i`/2−Θ2i`/2) · y`.

Since yi = 1 and y` = 0, the above inequality is equivalent to

sin(γi` + Θ1i`/2 + Θ

2i`/2) > 0.

Recalling the assumption that Θ1i` and Θ2i` are in the monotone

regime [−γi`, γi`], one can write

γi` + Θ1i`/2 + Θ

2i`/2 ∈ [0, 2γi`] ⊂ [0, π].

Hence, it is enough to show that

γi` + Θ1i`/2 + Θ

2i`/2 ∈ (0, 2γk`) ⊂ (0, π).

If γi` + Θ1i`/2 + Θ2i`/2 = 0, then it holds that

Θ1i` = Θ2i` = −γi`.

This contradicts the inequality ∆i` = Θ1i` −Θ2i` > 0. If γi` +Θ1i`/2 + Θ

2i`/2 = 2γk`, then it holds that

Θ1i` = Θ2i` = γi`,

which also contradicts the inequality ∆i` > 0. Combining thetwo cases, we obtain that sin(γi` + Θ1i`/2 + Θ

2i`/2) > 0 and

the inequality

sin(γi` + Θ1i`/2 + Θ

2i`/2) · yi> sin(γi` −Θ1i`/2−Θ2i`/2) · y`.

holds strictly. It follows that y = ei satisfies the conditions instatement 2 of Lemma 1.

C. Proof of Corollary 4

Proof. We only prove the strong uniqueness part since theproof for weak uniqueness is similar. Suppose that Θ1 andΘ2 are two solutions to problem (1) in the monotone regimesuch that Θ2 ∈ N (G,Θ1,W). Using the results of Theorem3, we only need to show that the induced orientation ofΘ1 − Θ2 is not weakly feasible. Assume conversely thatthe induced orientation A is a weakly feasible orientation.Then, by hypothesis, there exists a directed cycle (k1, . . . , kt)containing at least one normal edge such that∑

kiki+1 is normal

ωkiki+1 < 2π, (7)

where kt+1 := k1. We denote ∆k` := Θ1k`−Θ2k` and it followsthat

0 < ∆kiki+1 ≤ ωkiki+1 ∀i s.t. {ki, ki+1} is normal, (8)∆kiki+1 = 0 ∀i s.t. {ki, ki+1} is not normal,

where the right part of the first inequality is because Θ2 ∈N (G,Θ1,W). Combining inequalities (7) and (8) yields that

0 <

t∑i=1

∆kiki+1 =∑

kiki+1 is normal

∆kiki+1 (9)

≤∑

kiki+1 is normal

ωkiki+1 < 2π.

However, by the definition of ∆k` and Θk`, one can writet∑i=1

∆kiki+1 =

t∑i=1

Θ1kiki+1 −t∑i=1

Θ2kiki+1

=

t∑i=1

[Θ1ki −Θ

1ki+1

]−

t∑i=1

[Θ2ki −Θ

2ki+1

]= 0,

where the second last equality is the congruence relationmodule 2π and the last equality is because (k1, . . . , kt) isa cycle. This contradicts equation (9). Thus, the inducedorientation is not a weakly feasible orientation and the stronguniqueness holds.

D. Proof of Theorem 6

Proof. To prove the first inequality, we only need to notice thatany feasible orientation is also a weakly feasible orientationand the size of eye is equal to the girth when all edges arenormal.

Then, we consider the second inequality. Assume converselythat the maximal eye is attained by a directed cycle withchords in the weakly feasible orientation A. Without loss of

15

generality, assume that the directed cycle (1, . . . , t) attainsthe maximal eye with fewest chords, where t ≥ e(G) and{1, i} ∈ E is a chord for some i ∈ {3, . . . , t−1}. We considerfour different cases:

1. A1,i = 0: Consider the directed cycle

(1, i, i+ 1, . . . , t),

which has at most e(G) normal edges and strictly fewerchords than (1, . . . , t). This contradicts the assumptionthat the cycle (1, . . . , t) is a directed cycle that attainsthe size of eye with fewest chords.

2. A1,i = +1: and there exists at least one normal edgeamong {1, 2}, . . . , {i− 1, i}: The directed cycle

(1, i, i+ 1, . . . , t)

has at most e(G) normal edges and strictly fewer chordsthan (1, . . . , t). This also contradicts the assumption on(1, . . . , t).

3. A1,i = +1 and edges {1, 2}, . . . , {i−1, i} are not normal:Consider the directed cycle

(1, i, i− 1, . . . , 2),

which has exactly one normal edge and strictly fewerchords. By the definition of the maximal eye, we knowe(G) ≥ 1 and the cycle (1, i, i − 1, . . . , 2) has at moste(G) ≥ 1 normal edges. Hence, this contradicts theassumption on (1, . . . , t).

4. A1,i = −1: Consider the orientation à defined as

Ãk` := −Ak` ∀{k, `} ∈ E

and use the discussion in the first three cases.Combining the above four cases concludes that the maximaleye of the power network G must be attained by a chordlesscycle. Hence, the maximal eye is upper bounded by the longestchordless cycle.

E. Proof of Lemma 7

Proof. By the definition of strong uniqueness and weakuniqueness, if a solution to problem (1) is strongly unique,than it is also weakly unique. We only need to considerthe other direction. Assume conversely that there exists asolution Θ1 in the monotone regime that is weakly uniquebut not strongly unique. Then, there exists another solutionΘ2 ∈ N (G,Θ1,W) that is different from Θ1 accordingto Definition 5. Then, the phase difference of some line isdifferent for the two solutions. Considering the power injectionbalance at each bus, we know that the phase difference isdifferent at all lines. This means that the two solutions Θ1 andΘ2 are different according to Definition 4, which contradictsthe assumption that Θ1 is weakly unique.

F. Proof of Theorem 8

Proof. The sufficient part is proved in Corollary 4 and we onlyprove the necessary part. In this proof, bus n + 1 is definedas bus 1. We assume that

n∑i=1

ωi,i+1 ≥ 2π

We construct a power network G ∈ G and power injectionP such that there exist two different solutions Θ1,Θ2 in themonotone regime and Θ2 ∈ N (G,Θ1,W). Without loss ofgenerality, assume that

n∑i=1

ωi,i+1 = 2π.

This is because the construction for

W̃ :=

{2π∑n

j=1 ωj,j+1· ωi,i+1 : i ∈ [n]

}.

also works for the original W = {ωi,i+1 : i ∈ [n]} if∑nj=1 ωj,j+1 ≥ 2π. We define two phase angle vectors as

Θ11 := 0, Θ1i :=

i∑j=2

ωj,j+1 ∀i ∈ {2, . . . , n},

Θ2i := 0 ∀i ∈ [n].

Then, it follows that

Θ1i,i+1 = ωi,i+1, Θ2i,i+1 = 0 ∀i ∈ [n],

which means that Θ1 and Θ2 are both in the monotone regime.Since ωi,i+1, γi,i+1 ∈ (0, π/2], we know that γi,i+1 +ωi,i+1 ∈(0, π] and therefore, by the monotonicity of cos(·) in [0, π],we have

cos(γi,i+1 + ωi,i+1) < cos(γi,i+1).

For each line {i, i+ 1}, we define the positive constant

Ci,i+1 := |vivi+1|−1[− cos(γi,i+1 + ωi,i+1) + cos(γi,i+1)]−1

and the complex admittance

Bi,i+1 := sin(γi,i+1)Ci,i+1, Gi,i+1 := cos(γi,i+1)Ci,i+1.

We use p̃i,i+1(Θ) to denote the real power flow from bus ito bus i + 1 given the phase angle vectors Θ. Then, we cancalculate that

p̃i,i+1(Θ1)− p̃i,i+1(Θ2)

= −Gi,i+1|vivi+1|[cos(Θ1i,i+1)− cos(Θ2i,i+1)]+Bk`|vivi+1|[sin(Θ1i,i+1)− sin(Θ2i,i+1)]

= − cos(γi,i+1)Ci,i+1|vivi+1|[cos(ωi,i+1)− 1]+ sin(γi,i+1)Ci,i+1|vivi+1|sin(ωi,i+1)

= Ci,i+1|vivi+1|·[− cos(γi,i+1 + ωi,i+1) + cos(γi,i+1)]= 1.

It follows that

P̂i(Θ1)− P̂i(Θ2)

=[p̃i−1,i(Θ

1)− p̃i−1,i(Θ2)]−[p̃i,i+1(Θ

1)− p̃i,i+1(Θ2)]

=1− 1 = 0.

If we choose P := P̂ (Θ1), then Θ1 and Θ2 are two differentsolutions to problem (1) in the monotone regime such thatΘ2 ∈ N (G,Θ1,W) and that the strong uniqueness does nothold.

16

G. Proof of Lemma 10

Proof. For the notational simplicity, we denote the maximaleye and the maximal girth of the graph (V,E,W ) as e andg, respectively. Since the graph is 2-vertex-connected, theredoes not exist a degree-1 vertex. By Lemmas 13 and 15,Type II Operations do not change the maximal eye and themaximal girth of the graph. Moreover, the graph has a nestedear decomposition {L0, L1, . . . , Lr−1} by Theorem 9. Hence,we can assume that there is no degree-2 vertex except the slackbus. Assume conversely that graph (V,E,W ) is the 2-vertex-connected SP graph with minimal number of ears such thate > g. We will show that there must exist another graph withfewer ears in the ear decomposition and e > g. This will leadto a contradiction with our assumption that this graph has theminimal number of ears. If the graph has at most two ears,then the graph is a single line of a cycle and we know e = g.Hence, there exist at least three ears in the graph (V,E,W ).

a) Step 1: In this step, we prove that the graph has apair of parallel edges that contains a leaf ear, which we willdefine below. Since a nested ear decomposition is also a treedecomposition, we can assign a directed tree structure to earsin the decomposition. Here, we call an ear Lk a descendantear of L` if Lk is a descendant node of L` on the directedtree, or equivalently, both endpoints of ear Lk are on L` andat least one of them is different from the endpoints of L`. Wealso call ear L` the precedent ear of Lk. For any ear L`,we say that ear Lk is a smallest descendant ear of L` ifLk is a descendant ear of L` and there does not exist anotherear Li such that Li is also a descendant ear of L` and theinterval formed by the endpoints of Li on L` is a strict subsetof the interval formed by the endpoints of Lk. We note thateach ear may have multiple smallest descendant ears. We saythat an ear is a leaf ear if it is the smallest descendant earof some ear and has no descendant ear. We denote the set ofleaf ears as L. Considering the directed tree structure of theear decomposition, we know that the set L is not empty.

Suppose that Lk is a leaf ear with the endpoints k1, k2 andthat L` is the precedent ear of Lk. Since we have deletedall degree-2 vertices except the slack bus, ear Lk is eithera single line {k1, k2} or two edges {k1, k3} and {k2, k3}connecting the endpoints to the slack bus k3. Similarly, thepath connecting the two endpoints of Lk on the precedentear L`, which we denote as Pk, is either a single line orcontains the slack bus. Considering the ear Lk and the pathPk, there are two cases: two parallel edges with endpoints{k1, k2}, or one is a single line and the other is two edgeswith the slack bus. If the first case occurs, we have a pairof parallel edges containing a leaf ear. Now, we consider thesecond case. If we exchange the two paths, i.e., let Pk be a leafear and Lk be a path on the precedent ear, then the structure ofnested ear decomposition is not changed. Hence, without lossof generality, assume that Lk is a single line and Pk containsthe slack bus. If there exists an ear Lj different from L` thatalso contains leaf ears, then by the uniqueness of slack bus,the first case occurs for leaf ears on ear Lj .

Hence, we simply need to consider the case when L` is theonly ear that contains leaf ears. We consider the root ear L0.

By the definition of tree ear decomposition, we know that L0is a single line; let `1, `2 be the two endpoints of L0. Sinceall vertices except the slack bus have degree at least 3 andthe slack bus is not an endpoint of ears, both `1 and `2 havedegree at least 3. This implies that the root ear L0 has at least 2descendant ears and all descendant ears have endpoints `1, `2.Let Lk1 , Lk2 , . . . , Lkm be the descendant ears of L0. For eachLki , we define a sub-graph of (V,E,W ) consisting of ear L0and ears that are descendant nodes of Lki in the directed treeof ears. We can verify that each sub-graph also has a nestedear decomposition and therefore contains at least one leaf ear,which implies that ear L` belongs to all sub-graphs. On theother hand, due to the tree structure, the intersection of twodifferent sub-graphs is ear L0 and is not a leaf ear. Hence, theleaf ears in different sub-graphs are different and L` = L0. Itfollows that all descendant ears of L0 are leaf ears and theyform at least a pair of parallel edges containing a leaf ear.

b) Step 2: In this step, we construct a nested ear decom-position of the graph (V,E,W ) such that there exists a pairof parallel edges that contains the root ear L0 and that alledges are ears in the ear decomposition. According to Step 1,there exists a pair of parallel edges that contains a leaf ear.We denote the leaf ear in the pair of parallel edges as Lk. Weconsider the (undirected) cycle containing L0 and Lk. Supposethat the cycle has a non-empty edge intersection with earsLk0 , . . . , Lkt , where k0 = 0, kt = k and Lks+1 is a descendantear of Lks for s = 0, 1, . . . , t − 1. Notice that the endpointsof each ear Lks are on the cycle. Now, we construct a newnested ear decomposition L̃0, . . . , L̃m−1 such that Lk = L̃0is the root ear. We define L̃0 := Lk and L̃k as the remainingpart of the cycle. For ears Lks with 1 ≤ s ≤ t− 1, we defineL̃ks as the ear Lks with edges on the cycle deleted. For earsthat do not intersect with the cycle, we define L̃i := Li. It isdesirable to show that with the new set of ears still forms anested ear decomposition. To this end, we analyze three cases:• Case I. First, it can be verified that ears L̃k1 , . . . , L̃kt−1

are nested ears on L̃kt . Hence, ears L̃k0 , . . . , L̃kt stillform a nesting structure.

• Case II. Next, we consider an ear L̃i = Li that is notchanged and has both endpoints on Lks for some s ∈{0, 1, . . . , t−1}. Since Lks+1 is a descendant ear on Lks ,by the definition of nested ear decomposition, we knowthat the endpoints of L̃i are either both on L̃ks or bothon L̃kt . For the first case, Li is an ear on L̃ks and earson L̃ks have the same nesting structure as Lks . For thesecond case, both endpoints of Lk locate on L̃kt and arenested between the endpoints of L̃ks and L̃ks−1 . We notethat for the case when s = 0, both endpoints are equal tothe endpoints of L0 and they form the smallest possibleinterval on L̃kt . Hence, ears on L̃kt also have a nestedstructure.

• Case III. Finally, we consider ears that are not changedand do not have endpoints on Lks for any s = 0, . . . , t.These ears still form a nested structure on the originalprecedent ear and the nested ear decomposition structureis not changed.

Combining the above three cases concludes that the new setof ears is also a nested ear decomposition. Moreover, the

17

topological structure of the graph is not changed. Hence, inthe new ear decomposition, the root ear L̃0 = Lk has paralleledges. Finally, we observe that the parallel edges of the rootear are also ears in the ear decomposition.

c) Step 3: Suppose that the maximal eye is achieved bythe weakly feasible orientation A. In this step, we show that wecan modify A such that each edge with direction 0 is inciden