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Embeddings of Cubic Halin GraphsA Surface-by-Surface Inventory
Jonathan L. Gross
Columbia University, New York, NY 10027
Abstract
We derive an O(n2)-time algorithm for calculating the sequence ofnumbers g0(G), g1(G), g2(G), ... of distinct ways to embed a 3-regularHalin graph G on the respective orientable surfaces S0, S1, S2, ... .Key topological features are a quadrangular decomposition of planeHalin graphs and a new recombinant-strands reassembly process thatfits pieces together three-at-a-vertex. Key algorithmic features arereassembly along a post-order traversal, with just-in-time dynamic as-signment of roots for quadrangular pieces encountered along the tour.
1. Introduction
2. Some preexisting results
3. Quadrangulating a plane Halin graph
4. Reassembling from quadrangles
5. Partials and productions
6. Productions for Halin graphs
7. Sample calculation
8. Conclusions and open problems
* [email protected]; http://www.cs.columbia.edu/gross/.
1
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Embeddings of Cubic Halin Graphs: An Inventory 2
1 Introduction
We begin with a quick explanation of the title.
Halin graphs
A Halin graph is constructed as follows:
1. Let T be a plane tree
with at least 4 vertices no vertices of degree 2
2. Draw a cycle thru the leaves of the tree
in the order they occur in a preorder traversal.
Figure 1.1: A Halin graph for a 14-vertex tree with 8 leaves.
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Embeddings of Cubic Halin Graphs: An Inventory 3
Inventories of Embeddings
Def. The genus distribution for graph G is the sequence
dist(G) : g0(G), g1(G), g2(G),
EXAMPLE: K2 K3 a.k.a. CL3
Figure 1.2: g0 = 2, g1 = 38, g2 = 24
Notation Si denotes the orientable surface of genus i.
Notation gi(G) denotes # embeddings G Si.
Notation min(G) is the minimum genus of G.
Notation max(G) is the maximum genus of G.
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Embeddings of Cubic Halin Graphs: An Inventory 4
Counting Embeddings
Def. Let G be a labeled graph and S an oriented surface. Theembeddings : G S and : G S are equivalent iff thereexists an autohomeomorphism h : S S that
maps (G) isomorphically to (G);
preserves every label of G;
preserves the orientation of S.
2 1 3
0
2 13
0
Figure 1.3: Looking alike is not enough.
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Embeddings of Cubic Halin Graphs: An Inventory 5
Graph Amalgamations
In general, amalgamating two graphs means identifying asubgraph in one of them to an isomorphic subgraph in the other.
Vertex-amalgamation and edge-amalgamation are thetwo simplest kinds of amalgamation of two graphs.
=*V
Figure 1.4: Amalgamation of two graphs at a vertex.
=*E
Figure 1.5: Amalgamation of two graphs on an edge.
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Genus Distributions of Graph Amalgamations
=u v
*
Figure 1.6: Bar-amalgamation of two graphs.
dist(G H) = deg(u) deg(v) dist(G) dist(H)
here means convolution; [GrFu87]
g-dist of vertex amalg [GrKhPo10, KhPoGr10, Gr11].
g-dist of edge-amalg [PoKhGr10a, PoKhGr10b].
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2 Some Preexisting Results
About Minimum Genus
Thm 2.1 Calculating min(G) is NP-hard. [Th89]
min(G) is known for relatively few graphs:
complete graphs Kn [RiYo68]
complete bipartite graphs Km,n [Ri65]
hypercubes Qn [Ri55]
a bunch of less prominent families of graphs
K6 K3,4 Q3
Figure 2.1: Three families solved by Ringel.
Remark By way of contrast, there are combinatorial characterizations of
maximum genus by [Xu79] and [Ne81] and a polynomial-time algorithm by[FuGrMc88]. There are maximum genus determinations (e.g., see [Sk91])for families of graphs for which minimum genus calculation would seem tobe formidably hard. An up-to-date overview of maximum genus is given by[ChHu09].
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Minimum Genus of Graph Amalgamations
Thm 2.2 (Additivity of minimum genus)
min(G V H) = min(G) + min(H)
Proof [BHKY62]
By way of contrast, consider the edge-amalgamation K5EK5.
Figure 2.2: Amalgamation of two copies of K5 on an edge.
Obviously, the minimum genus is at least 1 and at most 2.
But is it really 2?
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NO! min(K5 E K5) = 1.
Y 0 2 X 1 Y
20Y1X2
K5
-> S1
Y 0 2 X 1 Y
20Y1X2
K5
-> S1
Y 0 2 X 1 Y
20Y1X2
K5 *E
K5
-> S1
Figure 2.3: K5 E K5 S1
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About Genus Distributions
Calculating dist(G) requires calculating min(G). The graphswhose genus distributions are known are planar or one edge awayfrom planarity.
Closed-end ladders [FuGrSt89] (derived 1984).
Figure 2.4: The closed-end ladder L4.
Circular ladders and Mobius ladders [McG87] (derived 1986).
Figure 2.5: Circular ladder CL4. and Mobius ladder M L4.
Ringel ladders [Te00].
Figure 2.6: Ringel ladder RL4.
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Some g-dist deriviations use [Ja87] (group representations).
Bouquets [GrRoTu89].
Figure 2.7: Bouquets B1, B2, and B3.
Dipoles [Ri90], [KwLe93].
Figure 2.8: Dipoles D1, D2, and D3.
Fans [ChMaZo11].
Figure 2.9: Fans F3 and F5.
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Cubic outerplanar graphs [Gr11].
Figure 2.10: A cubic outerplanar graph.
4-regular outerplanar graphs [PoKhGr11].
Figure 2.11: A 4-regular outerplanar graph.
Remark A survey of genus distributions, including average genus, is givenby [Gr09].
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3 Quadrangulating a Plane Halin Graph
We decompose a plane Halin graph H S0 into atomic partsthat will facilitate a genus distribution calculation.
We regard the vertices and the edges of the Halin graph itselfas black.
Step 1. In each edge on the exterior cycle, insert a red midpoint.
Figure 3.1: Halin graph with red midpoints on exterior cycle.
Since H is a Halin graph, there is exactly one exterior edgeon each polygonal face of the plane embedding.
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Step 2. Join each red vertex v to all of the non-leaf vertices on
the boundary of the face in whose boundary v lies.
Figure 3.2: Halin graph with all of its red edges.
Prop 3.1 The red and black edges together triangulate the re-gion inside the exterior cycle of the Halin graph.
Prop 3.2 Every black tree edge lies on two of the triangles formedby Steps 1 and 2.
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Step 3a. For each black tree edge, we pair the two incident
triangles into a quadrangle.Step 3b. We assign (unseen) colors blue, green, and brown theto tree edges so as to form a proper edge 3-coloring.
Step 3c. We visibly color each quadrangle with the unseencolor of the tree edge that bisects it.
Figure 3.3: Quadrangulation of a plane Halin graph.
Remark Any tree of maximum degree 3 is edge-3-colorable(via greedy algorithm).
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Step 4. Separate the quadrangulated map into quadrangles.
Figure 3.4: Separated quadrangles of a plane Halin graph.
Remark This quadrangulation works for any Halin graph, re-gardless of the degrees of its vertices.
Whats Next??
4 a puzzle: reassembling quadrangles into the Halin graph.
5 how this leads to a genus distribution.
6 the new set of productions for Halin graphs.
7 application to our running example of a Halin graph.
8 conclusions and open problems
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4 Reassembling from Quadrangles
u
w
v x
z
y
Quadrangulation puzzle for a plane cubic Halin graph H S0:
1. Each quadrangle Q is regarded as an initial fragment.
2. An RR-path on a fragment boundary is a
2-path with two red edges,
from a red vertex
through a black vertex
to another red vertex.
3. Initially, all RR-paths are said to be live.
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4. A legal move is initiated by choosing a vertex v such that
v is previously unchosen;
at least one fragment at v is a quadrangle;
all three RR-paths through v are live RR-paths.
Then the three fragments that meet at v are merged into asingle (larger) fragment.
If there is more than one live RR-path on the boundaryof the merged fragment, then all but one of the live RR-
paths are deemed to be dead.
5. You LOSE if you run out of legal moves before the map isfully reassembled.
This happens whenever there occurs an unmerged vertex wsuch that
there is a dead RR-path through w;
none of the fragments meeting at w is a quadrangle.
6. You WIN the game by reassembling the plane map.
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Attempt 1. Start with a merger at v.
Analysis. There are three live RR-paths on the boundary ofthe merged fragment.
u
w
vx
z
y
live RR-path
Figure 4.1: Attempt #1.
Result. LOSE, because RR-paths through two of the unmergedvertices u,w,x become dead.
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Attempt 2. First choose u and then choose v.
Analysis. There are two live RR-paths on the boundary of themerged fragment.
u
w
vx
z
y
live RR-path
Figure 4.2: Attempt #2.
Result. LOSE, because the RR-path through one of the un-merged vertices w, x becomes dead.
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Attempt 3. Start with u , w , y, z.
u
w
v x
z
y
Figure 4.3: Attempt #3.
Result. LOSE, since after there is a merger at v or x, there willbe no quadrangle at the remaining unmerged vertex.
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Solution: post-order traversal
post-order traversal for a plane tree: trace the boundary ofthe only region and call out the name of a vertex whenever yousee it for the last time.
u y
v x
w z
Figure 4.4: Post-order traversal.
z y x u v w
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u
w
v x
z
y
1. As a root for the inscribed tree of the Halin graph, chooseany leaf-vertex. (Must be a leaf to win.)
2. Choose vertices in the order in which they occur on a post-order traversal of the tree.
SOLUTION: for a root at the lower left corner
SOLUTION: z y x u v w
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z y x u v w
u
w
v x
z
y u
w
v x
z
y
u
w
vx
z
y
u
w
vx
z
y
u
w
v x
z
y u
w
v x
z
y
Figure 4.5: Solving the puzzle with post-order traversal.
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5 Partials and Productions
Partial Genus Distributions
In a single-rooted graph (G, r) with deg(r) = 2, the root r lieseither on two distinct fb-walks (type di) or twice on the same fb-walk (type si). We partition gi(G, r) into single-root partials:
di(G, r) = the number of embeddings of type-di, and
si(G, r) = the number of embeddings of type-si.
type-d0 type-s1
rr
Figure 5.1: The types of single-root partials for a 2-valent root.
Example 5.1
d0(B2, r) = 4 d1(B2, r) = 0s0(B2, r) = 0 s1(B2, r) = 2
g0(B2, r) = 4 g1(B2, r) = 2
Example 5.2
d0(K4, r) = 2 d1(K4, r) = 8s0(K4, r) = 0 s1(K4, r) = 6
g0(K4, r) = 2 g1(K4, r) = 14
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Double-Rooted Graphs
When two single-rooted graphs are pasted together acrosstheir single roots, there is no surviving root. However, if twodouble-rooted graphs are pasted together with one root of onegraph matched to one root of the other graph, then the mergedgraph has two roots.
(G,b,c) H(d, e) (X,b,e)
This permits iterated amalgamations.
Figure 5.2: An iterated edge-amalgamation.
Example 5.3 By following iterated amalgamation with a self-amalgamation, we can obtain graphs like circular ladders andMobius ladders.
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Whereas a single-edge-rooted graph (G, c) has only two par-
tials, namelydi(G, c) and si(G, c)
a double-edge-rooted graph (G,c,d) has the four partials
ddi(G,c,d), dsi(G,c,d), sdi(G,c,d), and ssi(G,c,d)
Example 5.4 (double-root partial g-dist for K4)
Figure 5.3: K4 with two roots at midpoints of two edges.
i ddi dsi sdi ssi gi0 2 0 0 0 21 4 4 0 2 14
Table 5.1: Partial genus distribution for (K4, u , v).
Remark The number of partials increases exponentiallyaccording to the number of roots. It follows that the num-ber of roots employed by a method of genus distribution calcu-lation must be limited.
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There are three sub-partials within the dd-type:
dd the two roots sharing two fb-walks;
dd the two roots sharing one fb-walk;
dd0 no shared fb-walks for the two roots.
type dd0
type dd'
type dd"
Figure 5.4: The three types of dd-subpartials.
Remark There are 10 double-root sub-partials for two edge-roots that both have 2-valent endpoints.
Remark The number of partials increases exponentially(also) with the degrees of vertex-roots and with degreesof the endpoints of the edge-roots, corresponding to a largernumber of possible configurations of fb-walks at the root.
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Deriving Productions with Recombinant Strands
Example 5.5
ddi (G,b,c) ddj (H,d,e) 4dd
i+j(X,b,e) + 2ss
2i+j+1(X,b,e)
b
e
b
e
b
e
b
e
b
eb
b
c
d
e
edd"i
dd"j
dd'i+j
dd'i+j dd'i+j dd'i+j
ss2i+j ss2i+j
Figure 5.5: Derivation of a production for vertex-amalgamation.
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6 Productions for Halin Graphs
We now introduce a new form of graph amalgamation, in whichseveral graphs are merged in a cycle around a vertex.
For cubic Halin graphs, we merge three graphs at a time,exactly as for the puzzle, so that one of them is a quadrangleQ = K4 e.
A
Q
X
B
vv
vv
r
s
z
y
s'
t'
t
r'
Figure 6.1: A three-way pie-merge (A , B , Q) X.
Prop 6.1 In a pie-merge (A , B , Q) X, each rotation system for X is consistent with exactly two rotation systems for A and
exactly two for B.
Remark In this oral presentation, we use the picture to definea pie-merge.
Remark A series of fundamental papers ([GrKhPo10], [Gr10a], [KhPoGr10],[PoKhGr10a], [PoKhGr10b], and [Gr10b]) derives the productions that cor-respond to various ways of synthesizing graphs from graphs whose partialgenus distributions are known.
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Six Relevant Partials
For Halin graphs, we split gi into six double-rooted partials:
ss1 ss2sd'ds'dd"dd'
u uuuuu
v v vvvv
Figure 6.0: The 6 double-rooted partials for 3-way pie-merge.
Here is what they count:
dd Each of the roots u and v lies on two distinct fb-walks. Oneand only one of these fb-walks traverses both roots.
dd Each of the roots u and v lies on two distinct fb-walks. Bothof these fb-walks traverse both roots.
ds Root u lies on two distinct fb-walks. One of these fb-walkstraverses root v twice.
sd Root v lies on two distinct fb-walks. One of these fb-walkstraverses root u twice.
ss1 A single fb-walks traverses roots u and v twice. The occur-rences of each root are consecutive.
ss2 A single fb-walks traverses roots u and v twice. The occur-rences of the two roots alternate.
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Computational Complexity
Thm 6.2 (a) The values of the six partials for the amalga-mands A and B completely determine those for the result
X of a pie-merge (A , B , Q) X.
(b) For |VA| = k and |VB| = m, there is an O(km)-time al-gorithm for calculating the partial genus distribution of the
resulting graph X of a pie-merge (A , B , Q) X.
Cor 6.3The post-order traversal using the 36 productions cor-
responding to the six partials yields an O(n2) algorithm for thegenus distribution of a cubic Halin graph with n vertices.
Practical Calculation with Spreadsheet
Express each of the six partial genus distributions of X as asum of convolutions of partials of A and B, as prescribed by theproductions. (See 7.)
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Thirty-Six Productions
1. ddi ddj dd
i+j + 2dd
i+j+1 + ss
2i+j+1.
2. ddi ddj 2dd
i+j + 2ss
2i+j+1.
3. ddi dsj 2dd
i+j + 2ss
2i+j+1.
4. ddi sdj 2sd
i+j + 2ss
1i+j+1.
5. ddi ss1j 4sd
i+j.
6. ddi ss
2j 2ds
i+j + 2sd
i+j.
7. ddi ddj 2dd
i+j + 2ss
2i+j+1.
8. ddi ddj 4dd
i+j.
9. ddi dsj 4ds
i+j.
10. ddi sdj 4sd
i+j.
11. ddi ss1j 4ss
1i+j.
12. ddi ss2j 2dd
i+j1 + 2ss
2i+j.
13. dsi ddj 2ds
i+j + 2ss
1i+j+1.
14. dsi ddj 4ds
i+j.
15. dsi dsj 4ds
i+j.
16. dsi sdj 4ss
1i+j.
17. dsi ss1j 4ss1i+j.
18. dsi ss2j 2ds
i+j1 + 2ss
1i+j.
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19. sdi ddj 2dd
i+j + 2ss
2i+j+1.
20. sdi ddj 4sd
i+j.
21. sdi dsj 2dd
i+j1 + 2ss
2i+j.
22. sdi sdj 4sd
i+j.
23. sdi ss1j 4sd
i+j1.
24. sdi ss2j 2dd
i+j1 + 2ss
2i+j.
25. ss1
i dd
j 4ds
i+j.26. ss1i dd
j 4ss
1i+j.
27. ss1i dsj 4ds
i+j1.
28. ss1i sdj 4ss
1i+j.
29. ss1i ss1j 4ss
1i+j1.
30. ss1i ss2j 4ds
i+j1.
31. ss2i ddj 2ds
i+j + 2sd
i+j.
32. ss2i ddj 2dd
i+j1 + 2ss
2i+j.
33. ss2i dsj 2dd
i+j1 + 2ss
2i+j.
34. ss2i sdj 2sd
i+j1 + 2ss
1i+j.
35. ss2i ss1j 4sd
i+j1.
36. ss2i ss
2j 2dd
i+j1 + dd
i+j2 + ss
2i+j1.
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Derivations of Pie-Merge Productions
Figure 6.1: Prod #1: ddi ddj dd
i+j + 2dd
i+j+1 + ss
2i+j+1.
Figure 6.2: Prod #18: dsi ss2j 2ds
i+j1 + 2ss
1i+j.
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7 Sample Calculation
Merger at z
Graph A (K4 e):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 0 0 0 0 2 2
Graph B (K4 e):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 0 0 0 0 2 2
Use Productions 1, 6, 31, and 36.
Merged Graph K4:
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 4 4 4 0 2 14
Merger at y
Just like the merger at z.
Merged Graph: K4
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 4 4 4 0 2 14
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Merger at x
Graph A (result from merger at z):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 4 4 4 0 2 14
Graph B (result from merger at y):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 4 4 4 0 2 14
Use 25 productions (all those without the partial ss1).
Merged Graph:
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 40 4 12 12 0 2 70
2 0 16 48 48 32 40 184
Merger at u
Just like the merger at z.
Merged Graph: K4
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 2
1 0 4 4 4 0 2 14
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Merger at v
Graph A (result from merger at x):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 40 4 12 12 0 2 702 0 16 48 48 32 40 184
Graph B (result from merger at u):
i dd
i dd
i ds
i sd
i ss1
i ss2
i gi0 2 0 0 0 0 0 21 0 4 4 4 0 2 14
Use 30 productions.
Merged Graph:
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 112 4 28 12 0 2 1582 544 96 544 352 80 112 17283 0 64 448 448 704 544 2208
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Merger at w
Graph A (result from merger at v):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 112 4 28 12 0 2 1582 544 96 544 352 80 112 17283 0 64 448 448 704 544 2208
Graph B (K4 e):
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 21 0 0 0 0 0 2 2
Merged Graph: final result
i ddi ddi ds
i sd
i ss
1i ss
2i gi
0 2 0 0 0 0 0 2
1 144 4 60 4 0 2 2142 1440 224 1632 224 56 144 37203 1024 1088 4800 1088 1088 1440 105284 0 0 0 0 896 1024 1920
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Embeddings of Cubic Halin Graphs: An Inventory 40
8 Conclusions and Open Problems
New features of this solution:
quadrangulation
pie-merge (invented for application to Halin graphs)
post-order traversal (also used in [Gr11] and [PoKhGr11])
dynamic assignment of roots to graphs
calculation for graphs of tree-width 3
Some open genus-distribution calculation problems:
1. wheels
2. planar cubic Hamiltonian graphs
3. truncated polyhedra
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Some High-Level Theoretical Questions
1. shape of genus distributions
unimodality of genus distributions
unimodality of the partials
approximation by Stirling cycle numbers
2. average genus (without calculating the entire g.d.)
The method presented here works for Halin graphs of boundedvalence. Halin graphs are known [Wi87] to be of treewidth 3.
Conjecture 8.1 There is a quadratic-time algorithm for thegenus distribution of any cubic graph of treewidth 3.
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Embeddings of Cubic Halin Graphs: An Inventory 42
References
[AHU83] A. V. Aho, J. E. Hopcroft, and J. D. Ullman, DataStructures and Algorithms, Addison-Wesley, 1983.
[BHKY62] J. Battle, F. Harary, Y. Kodama, and J.W. T. Youngs, Additivity of the genus of a graph,Bull. Amer. Math. Soc. 68 (1962), 565568.
[BWGT09] L. W. Beineke, R. J. Wilson, J. L. Gross, and T.W. Tucker (editors), Topics in Topological Graph Theory,
Cambridge University Press, 2009.
[ChHu09] J. Chen and Y. Huang, Maximum genus, Chapter 2 ofTopics in Topological Graph Theory(editors L. W. Beineke,R. J. Wilson, J. L. Gross, and T. W. Tucker), CambridgeUniversity Press, 2009.
[ChMaZo11] Y. Chen, T. Mansour, and Q. Zou, On the totalembeddings for some types of graphs, manuscript.
[FuGrMc88] M. Furst, J. L. Gross, and L. A. McGeoch, Findinga maximum-genus umbedding, J. ACM 35 (1988), 523534.
[FuGrSt89] M. L. Furst, J. L. Gross, and R. Statman, Genusdistribution for two classes of graphs, J. Combin. Theory(B) 46 (1989), 2236.
[Gr09] J. L. Gross, Distribution of embeddings, Chapter 3 ofTopics in Topological Graph Theory(editors L. W. Beineke,
R. J. Wilson, J. L. Gross, and T. W. Tucker), CambridgeUniversity Press, 2009.
[Gr10a] J. L. Gross, Genus distribution of graph amalgamations,II. Self-pasting at 2-valent co-roots, Preprint (2010), 21pp.
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[Gr10b] J. L. Gross, Genus distribution of graphs under surgery:
adding edges and splitting vertices, New York J. Math. 16(2010), 161178.
[Gr11] Genus distributions of cubic outerplanar graphs,preprint.
[GrFu87] J. L. Gross and M. L. Furst, Hierarchy for imbedding-distribution invariants of a graph, J. Graph Theory 11(1987), 205220.
[GrKhPo10] J. L. Gross, I. Khan, and M. Poshni, Genus distri-bution of graph amalgamations, I: Pasting two graphs at2-valent roots, Ars Combinatoria 94 (2010), 3353.
[GrKlRi93] J. L. Gross, E. W. Klein, and R. G. Rieper, Onthe average genus of a graph, Graphs and Combinatorics 9(1993), 153162.
[GrRoTu89] J. L. Gross, D. P. Robbins, and T. W. Tucker,
Genus distributions for bouquets of circles, J. Combin. The-ory (B) 47 (1989), 292306.
[GrTu87] J. L. Gross and T. W. Tucker, Topological Graph The-ory, Dover, 2001; original edn. Wiley, 1987).
[Ja87] D. M. Jackson, Counting cycles in permutations by groupcharacters, with an application to a topological problem,Trans. Amer. Math. Soc. 299 (1987), 785801.
[KhPoGr10] I. Khan, M. Poshni, and J. L. Gross, Genus distri-bution of graph amalgamations, III: Pasting when one roothas arbitrary degree, Ars Mathematica Contemporanea 3(2010), 121138.
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Embeddings of Cubic Halin Graphs: An Inventory 44
[KwLe93] J. H. Kwak and J. Lee, Genus polynomials of dipoles,
Kyungpook Math. J. 33 (1993), 115125.
[KwLe94] J. H. Kwak and J. Lee, Enumeration of graph embed-dings, Discrete Math. 135 (1994), 129151.
[McG87] L. A. McGeoch, Algorithms for two graph problems:computing maximum-genus imbedding and the two-server
problem, PhD thesis, Carnegie-Mellon University, 1987.
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Johns Hopkins University Press, 2001.
[Ne81] L. Nebesky, A new characterization of the maximumgenus of a graph, Czech. Math. J. 31 (1981), 604613.
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operation effect on genus distribution, II. Self-amalgamation on a pair of edges, Preprint (2010), 21 pages.
[PoKhGr11] M. Poshni, I. Khan, and J. L. Gross, Genus dis-tributions of 4-regular outerplanar graphs, Preprint (2011),22 pages.
[Ri90] R. G. Rieper, The enumeration of graph embeddings,Ph.D. thesis, Western Michigan University, 1990.
[Ri55] G. Ringel, Uber drei kombinatorische Prob-leme am n-dimensionalen Wurfel und Wurfelgitter,Abh. Math. Sem. Univ. Hamburg 20 (1955), 1019.
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Graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139150.
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9 Appendix
Derivations of the Productions
Figure 9.1: ddi ddj dd
i+j + 2dd
i+j+1 + ss
2i+j+1.
Figure 9.2: ddi ddj 2dd
i+j + 2ss
2i+j+1.
Figure 9.3: ddi dsj 2dd
i+j + 2ss
2i+j+1.
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Figure 9.4: ddi sdj 2sd
i+j + 2ss
1i+j+1.
Figure 9.5: ddi ss1j 4sd
i+j.
Figure 9.6: ddi ss2j 2ds
i+j + 2sd
i+j.
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Figure 9.7: ddi ddj 2dd
i+j + 2ss
2i+j+1.
Figure 9.8: ddi ddj 4dd
i+j.
Figure 9.9: ddi dsj 4ds
i+j.
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Figure 9.10: ddi sdj 4sd
i+j.
Figure 9.11: ddi ss1j 4ss
1i+j.
Figure 9.12: ddi ss2j 2dd
i+j1 + 2ss
2i+j.
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Figure 9.13: dsi ddj 2ds
i+j + 2ss
1i+j+1.
Figure 9.14: dsi ddj 4ds
i+j.
Figure 9.15: dsi dsj 4ds
i+j.
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Figure 9.16: dsi sdj 4ss
1i+j.
Figure 9.17: dsi ss1j 4ss
1i+j.
Figure 9.18: dsi ss2j 2ds
i+j1 + 2ss
1i+j.
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Figure 9.19: sdi ddj 2dd
i+j + 2ss
2i+j+1.
Figure 9.20: sdi ddj 4sd
i+j.
Figure 9.21: sdi dsj 2dd
i+j1 + 2ss
2i+j.
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Figure 9.22: sdi sdj 4sd
i+j.
Figure 9.23: sdi ss1j 4sd
i+j1.
Figure 9.24: sdi ss2j 2dd
i+j1 + 2ss
2i+j.
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Figure 9.25: ss1i ddj 4ds
i+j.
Figure 9.26: ss1i ddj 4ss
1i+j.
Figure 9.27: ss1i dsj 4ds
i+j1.
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Figure 9.28: ss1i sdj 4ss
1i+j.
Figure 9.29: ss1i ss1j 4ss
1i+j1.
Figure 9.30: ss1i ss2j 4ds
i+j1.
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Figure 9.31: ss2i ddj 2ds
i+j + 2sd
i+j.
Figure 9.32: ss2i ddj 2dd
i+j1 + 2ss
2i+j.
Figure 9.33: ss2i dsj 2dd
i+j1 + 2ss
2i+j.
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Figure 9.34: ss2i sdj 2sd
i+j1 + 2ss
1i+j.
Figure 9.35: ss2i ss1j 4sd
i+j1.
Figure 9.36: ss2i ss2j 2dd
i+j1 + dd
i+j2 + ss
2i+j1.
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