Characterizing Cohen-Macaulay power edge ideals oftrees

James Gossell

Clemson University

with M. Cowen, A. Hahn, W.F. Moore, S. Sather-Wagsta↵

October 21, 2020

James Gossell (Clemson) Power Edge Ideals October 21, 2020 1 / 16

Outline

James Gossell (Clemson) Power Edge Ideals October 21, 2020 2 / 16

Graph Theory Algebra

Electrical Engineering

1. Vertex Covers 2. Edge Ideals

3. Phasor Measurement Units (PMUs)

4. The PMU problem

6. MAIN THEOREM: Finding minimal PMU Covers

5. Power Edge Ideals

Motivation

Main Part of the Talk

Vertex Covers

Definition

Let G = (V ,E ) be a graph. A vertex cover of G is a subset V 0 ✓ V suchthat for each edge vivj in G either vi 2 V

0 or vj 2 V0. A vertex cover is

minimal if it does not properly contain another vertex cover.

Example

For the graph G = v1 v2

v4 v3

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 3 / 16

Vertex Covers

Definition

Let G = (V ,E ) be a graph. A vertex cover of G is a subset V 0 ✓ V suchthat for each edge vivj in G either vi 2 V

0 or vj 2 V0. A vertex cover is

minimal if it does not properly contain another vertex cover.

Example

For the graph G = v1 v2

v4 v3

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 3 / 16

Edge Ideals

Definition

Let G be a graph with vertex set V = {v1, . . . , vd}. The edge ideal of Gis the ideal IG ✓ R = A[X1, . . . ,Xd ] that is “generated by the edges of G”

IG = ({XiXj | vivj is an edge in G})R

Example

Set R = A[X1,X2,X3,X4]. The graph G = v1 v2

v4 v3

has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.We obtain the following irredundant m-irreducible decomposition:

IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .

James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16

Edge Ideals

Definition

Let G be a graph with vertex set V = {v1, . . . , vd}. The edge ideal of Gis the ideal IG ✓ R = A[X1, . . . ,Xd ] that is “generated by the edges of G”

IG = ({XiXj | vivj is an edge in G})R

Example

Set R = A[X1,X2,X3,X4]. The graph G = v1 v2

v4 v3

has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.We obtain the following irredundant m-irreducible decomposition:

IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .

James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16

Edge Ideals

Definition

Let G be a graph with vertex set V = {v1, . . . , vd}. The edge ideal of Gis the ideal IG ✓ R = A[X1, . . . ,Xd ] that is “generated by the edges of G”

IG = ({XiXj | vivj is an edge in G})R

Example

Set R = A[X1,X2,X3,X4]. The graph G = v1 v2

v4 v3

has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.

We obtain the following irredundant m-irreducible decomposition:

IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .

James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16

Edge Ideals

DefinitionThe edge ideal IG ✓ R has the following m-irreducible decompositions:

IG =\

V 0 min

PV 0

Example

Set R = A[X1,X2,X3,X4]. The graph G = v1 v2

v4 v3

has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.

We obtain the following irredundant m-irreducible decomposition:

IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .

James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16

Edge Ideals

DefinitionThe edge ideal IG ✓ R has the following m-irreducible decompositions:

IG =\

V 0 min

PV 0

Example

Set R = A[X1,X2,X3,X4]. The graph G = v1 v2

v4 v3

has edge ideal IG = (X1X2,X2X3,X2X4,X3X4)R .

The minimal vertex covers for G are {v1, v3, v4}, {v2, v3}, {v2, v4}.We obtain the following irredundant m-irreducible decomposition:

IG = (X1,X3,X4)R \ (X2,X3)R \ (X2,X4)R .

James Gossell (Clemson) Power Edge Ideals October 21, 2020 4 / 16

Vertex Covers and Edge Ideals

Theorem (R.H. Villareal)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed. (every minimal vertex cover has the same size)

(ii) G is a suspension of a subtree G0.

(iii) IG is Cohen-Macaualay.

Example

Let

T = v1,1 v1,2 v1,3

v2,1 v2,2 v2,3

, T0 = v1,1 v1,2 v1,3

Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16

Vertex Covers and Edge Ideals

Theorem (R.H. Villareal)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed. (every minimal vertex cover has the same size)

(ii) G is a suspension of a subtree G0.

(iii) IG is Cohen-Macaualay.

Example

Let

T = v1,1 v1,2 v1,3

v2,1 v2,2 v2,3

, T0 = v1,1 v1,2 v1,3

Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16

Vertex Covers and Edge Ideals

Theorem (R.H. Villareal)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed. (every minimal vertex cover has the same size)

(ii) G is a suspension of a subtree G0.

(iii) IG is Cohen-Macaualay.

Example

Let

T = v1,1 v1,2 v1,3

v2,1 v2,2 v2,3

, T0 = v1,1 v1,2 v1,3

Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16

Vertex Covers and Edge Ideals

Theorem (R.H. Villareal)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed. (every minimal vertex cover has the same size)

(ii) G is a suspension of a subtree G0.

(iii) IG is Cohen-Macaualay.

Example

Let

T = v1,1 v1,2 v1,3

v2,1 v2,2 v2,3

, T0 = v1,1 v1,2 v1,3

Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16

Vertex Covers and Edge Ideals

Theorem (R.H. Villareal)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed. (every minimal vertex cover has the same size)

(ii) G is a suspension of a subtree G0.

(iii) IG is Cohen-Macaualay.

Example

Let

T = v1,1 v1,2 v1,3

v2,1 v2,2 v2,3

, T0 = v1,1 v1,2 v1,3

Note that T is the suspension of T 0. Thus, T is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 5 / 16

PMU Covers and Edge Ideals

DefinitionIn an electrical power system, a bus is a substation wheretransmission lines meet.

Each line connects two buses.

Example

The graphv1 v2 v3 v4

v5 v6

represents a power system with six buses and six lines.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 6 / 16

PMU Covers and Power Edge Ideals

Definition

A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.

A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.

A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.

Example

v1

PMU

v2 v3 v4

PMU

v5 v6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16

PMU Covers and Power Edge Ideals

Definition

A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.

A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.

A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.

Example

v1PMU v2 v3 v4

PMU

v5 v6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16

PMU Covers and Power Edge Ideals

Definition

A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.

A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.

A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.

Example

v1PMU v2 v3 v4

PMU

v5 v6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16

PMU Covers and Power Edge Ideals

Definition

A Phasor measurement unit (PMU) is a device placed at a bus in anelectrical power system to monitor the voltage at the bus and thecurrent in all lines connected to it. A PMU placement is a set ofbuses where PMUs are placed.

A bus in the system is observable if its voltage is known. A line is thesystem is observable if its current is known.

A PMU cover of the system is a placement of some PMUs on busesmaking the entire system observable.

Example

v1PMU v2 v3 v4

PMU

v5 v6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 7 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

v

PMU

1

obs.

v

obs.

2

obs.

obs.

v

obs.

3

obs.

obs.

v

PMU

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

v

PMU

1

obs.

v

obs.

2

obs.

obs.

v

obs.

3

obs.

obs.

v

PMU

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.

v

obs.

2

obs.

obs.

v

obs.

3

obs.

obs.

v

PMU

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.v

obs.

2

obs.

obs.

v

obs.

3

obs.

obs.

v

PMU

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.

v

obs.

3

obs.

obs.

v

PMU

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.

v

obs.

3

obs.

obs.

vPMU4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.

v

obs.

3

obs.

obs.vPMU4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.

vobs.3

obs.

obs.vPMU4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.vobs.3

obs.

obs.vPMU4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.vobs.3

obs.

obs.vPMU4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.vobs.3

obs.

obs.vPMU4

vobs.5

obs.

vobs.6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to an observable vertex v are

observable except for one, then all of the lines incident to v are

observable.

Example

vPMU1

obs.vobs.2

obs.

obs.vobs.3

obs.

obs.vPMU4

vobs.5

obs.vobs.6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 8 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

v

obs.

1

obs.

vPMU2

obs.

obs.

v

obs.

3

obs.

obs.

v

obs.

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

v

obs.

1obs.

vPMU2

obs.

obs.v

obs.

3

obs.

obs.

v

obs.

4

v

obs.

5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

vobs.1

obs.vPMU2

obs.

obs.vobs.3

obs.

obs.

v

obs.

4

vobs.5

obs.

v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

vobs.1

obs.vPMU2

obs.

obs.vobs.3

obs.

obs.

v

obs.

4

vobs.5

obs.v

obs.

6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

vobs.1

obs.vPMU2

obs.

obs.vobs.3

obs.

obs.

v

obs.

4

vobs.5

obs.vobs.6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

vobs.1

obs.vPMU2

obs.

obs.vobs.3

obs.

obs.

v

obs.

4

vobs.5

obs.vobs.6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

vobs.1

obs.vPMU2

obs.

obs.vobs.3

obs.

obs.v

obs.

4

vobs.5

obs.vobs.6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Rule1 Place a PMU at vertex v . Then the vertex v is observable and every

line incident to v is observable.

2 (Ohm) A bus incident to an observable line is observable.

3 (Ohm) A line incident to two observable buses is observable.

4 (Kircho↵) If all the lines incident to a vertex v are observable except

for one, then all of the lines incident to v are observable.

Example

vobs.1

obs.vPMU2

obs.

obs.vobs.3

obs.

obs.vobs.4

vobs.5

obs.vobs.6

James Gossell (Clemson) Power Edge Ideals October 21, 2020 9 / 16

PMU Covers and Power Edge Ideals

Definition

Let G be a graph with vertex set V = {v1, . . . , vd} and setR = A[X1, . . . ,Xd ]. The power edge ideal of G is

IPG =

\

V 0

PV 0

where the intersection is taken over all PMU covers of G .

Example

v1 v2 v3 v4

v5 v6

IPG = hX1,X4i \ hX2i \ hX3i \ hX5i \ hX6i = hX1X2X3X5X6,X2X3X4X5X6i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 10 / 16

PMU Covers and Power Edge Ideals

Definition

Let G be a graph with vertex set V = {v1, . . . , vd} and setR = A[X1, . . . ,Xd ]. The power edge ideal of G is

IPG =

\

V 0

PV 0

where the intersection is taken over all PMU covers of G .

Example

v1 v2 v3 v4

v5 v6

IPG = hX1,X4i \ hX2i \ hX3i \ hX5i \ hX6i = hX1X2X3X5X6,X2X3X4X5X6i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 10 / 16

The PMU Placement Problem

The PMU Placement ProblemDetect any and all outages in the system and minimize cost.I.e., find the smallest number of PMUs needed to monitor the entiresystem.

A Problem with the PMU Placement Problem

The PMU Placement Problem is NP-complete (HHHH).

GoalStudy the problem algebraically and find some interesting examples ofideals along the way. Specifically, we will:

Characterize the trees T such that IPT is unmixed, i.e., such that allminimal PMU covers have the same size.

Characterize the trees T such that IPT is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 11 / 16

The PMU Placement Problem

The PMU Placement ProblemDetect any and all outages in the system and minimize cost.I.e., find the smallest number of PMUs needed to monitor the entiresystem.

A Problem with the PMU Placement Problem

The PMU Placement Problem is NP-complete (HHHH).

GoalStudy the problem algebraically and find some interesting examples ofideals along the way. Specifically, we will:

Characterize the trees T such that IPT is unmixed, i.e., such that allminimal PMU covers have the same size.

Characterize the trees T such that IPT is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 11 / 16

The PMU Placement Problem

The PMU Placement ProblemDetect any and all outages in the system and minimize cost.I.e., find the smallest number of PMUs needed to monitor the entiresystem.

A Problem with the PMU Placement Problem

The PMU Placement Problem is NP-complete (HHHH).

GoalStudy the problem algebraically and find some interesting examples ofideals along the way. Specifically, we will:

Characterize the trees T such that IPT is unmixed, i.e., such that allminimal PMU covers have the same size.

Characterize the trees T such that IPT is Cohen-Macaulay.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 11 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

Example

v1

PMU obs.

v2

obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs. obs.

v8

obs.

obs.

obs.

v9

PMU

v10

obs. obs.

v11

PMU obs.

v12

obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.

(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

Example

v1

PMU obs.

v2

obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs. obs.

v8

obs.

obs.

obs.

v9

PMU

v10

obs. obs.

v11

PMU obs.

v12

obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.

(iii) G is obtained by “nicely linking” a sequence of paths.-PMU Covers: Choose one vertex from each path

Example

v1

PMU obs.

v2

obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs. obs.

v8

obs.

obs.

obs.

v9

PMU

v10

obs. obs.

v11

PMU obs.

v12

obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU

obs.

v2

obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs. obs.

v8

obs.

obs.

obs.

v9PMU

v10

obs. obs.

v11PMU

obs.

v12

obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2

obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs. obs.

v8

obs.

obs.

obs.v9

PMU

v10

obs.

obs.v11

PMU obs.v12

obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7

obs.

obs.v8

obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6

obs.

obs.v7

obs. obs.v8

obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.

v3

obs. obs.

v4

obs.

v5

obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.v3

obs. obs.

v4

obs.

v5

obs.

obs.v6

obs. obs.v7

obs. obs.v8

obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.v3

obs.

obs.

v4

obs.

v5obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.v3

obs. obs.v4

obs.

v5obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-PMU Covers: Choose one vertex from each path

Example

v1PMU obs.

v2obs.

obs.

obs.v3

obs. obs.v4

obs.

v5obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-Generators of IPG : The product of the vertices in the paths.

Example

v1PMU obs.

v2obs.

obs.

obs.v3

obs. obs.v4

obs.

v5obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Characterizing Unmixed Trees

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:(i) G is unmixed with respect to PMU covers.(ii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iii) G is obtained by “nicely linking” a sequence of paths.

-Generators of IPG : The product of the vertices in the paths.

Example

v1PMU obs.

v2obs.

obs.

obs.v3

obs. obs.v4

obs.

v5obs. obs.

v6obs. obs.

v7obs. obs.

v8obs.

obs.

obs.v9

PMU

v10obs. obs.

v11PMU obs.

v12obs.

IPG = hX1X2X3X4,X5X6X7X8X9,X10X11X12i

James Gossell (Clemson) Power Edge Ideals October 21, 2020 12 / 16

Alegraic Implications

Summary

For any ideal I , we have

I is a CI =) I is Gor. =) I is CM =) I is unmixed

For any tree T , we have

IPT is a CI () I

PT is Gor. () I

PT is CM () I

PT is unmixed

ClaimIn general, for abitrary graphs G ,

IPG is a CI 6 I(=

P

G is Gor. 6 I(=P

G is CM 6 I(=P

G is unmixed

James Gossell (Clemson) Power Edge Ideals October 21, 2020 13 / 16

Alegraic Implications

Summary

For any ideal I , we have

I is a CI =) I is Gor. =) I is CM =) I is unmixed

For any tree T , we have

IPT is a CI () I

PT is Gor. () I

PT is CM () I

PT is unmixed

ClaimIn general, for abitrary graphs G ,

IPG is a CI 6 I(=

P

G is Gor. 6 I(=P

G is CM 6 I(=P

G is unmixed

James Gossell (Clemson) Power Edge Ideals October 21, 2020 13 / 16

Alegraic Implications

Summary

For any ideal I , we have

I is a CI =) I is Gor. =) I is CM =) I is unmixed

For any tree T , we have

IPT is a CI () I

PT is Gor. () I

PT is CM () I

PT is unmixed

ClaimIn general, for abitrary graphs G ,

IPG is a CI 6 I(=

P

G is Gor. 6 I(=P

G is CM 6 I(=P

G is unmixed

James Gossell (Clemson) Power Edge Ideals October 21, 2020 13 / 16

Summery

Theorem (Villareal)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed with respect to vertex covers.

(ii) G is a suspension of a subtree G0.

(iii) Every vertex of deg. � 2 is adjacent to exactly 1 vertex of deg. 1.(iv) IG is Cohen-Macaualay.

Theorem (CGHMS)

Let G be a tree. The following conditions are equivalent:

(i) G is unmixed with respect to PMU covers.

(ii) G is obtained by “nicely linking” a sequence of paths.

(iii) Every vertex of deg. � 3 is adjacent to exactly 2 vertices of deg. 2.(iv) I

PG is Cohen-Macaulay.

(v) IPG is Gorenstein and Complete Intersection.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 14 / 16

Counterexamples

Claim

IPG is a CI 6(= I

PG is Gor. 6(= I

PG is CM 6(= I

PG is unmixed

James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16

Counterexamples

Claim

IPG is a CI 6

?

(= IPG is Gor. 6(= I

PG is CM 6(= I

PG is unmixed

Example (IPG can be Gorenstein but not a Complete Intersection)

v1

v2 v3

v4 v7

v5 v8

v6 v9

v10 v11 v15 v13

v12 v14James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16

Counterexamples

Claim

IPG is a CI 6(= I

PG is Gor. 6

?

(= IPG is CM 6(= I

PG is unmixed

Example (IPG can be Cohen-Macaulay but not Gorenstein)

v1 v2 v3

v6 v5 v4

James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16

Counterexamples

Claim

IPG is a CI 6(= I

PG is Gor. 6(= I

PG is CM 6

?

(= IPG is unmixed

Example (IPG can be Unmixed but not Cohen-Macaulay)

v1 v2 v3 v4 v5

v6 v7 v8 v9 v10

James Gossell (Clemson) Power Edge Ideals October 21, 2020 15 / 16

References

Villareal, Rafael H., 1990

Cohen-macaulay graphs

Manuscripta Mathematica 66(1), 277-293.

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, Michael A.

Henning 2002

Domination in Graphs Applied to Electric Power Networks

SIAM J. Discrete Math 15(4), 519-529.

W. Frank Moore, Mark Rogers, Sean Sather-Wagsta↵ 2017

Monomial Ideals and Their Decompositions.

James Gossell (Clemson) Power Edge Ideals October 21, 2020 16 / 16