rooted minors and delta-wye transformations

97 Pages·2012·1.17 MB·English

ROOTED MINORS AND
DELTA-WYE TRANSFORMATIONS
by
Lino Demasi
B.Math., University of Waterloo, 2006
M.Math., University of Waterloo, 2008
a Thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
in the
Department of Mathematics
Faculty of Science
⃝c Lino Demasi 2012
SIMON FRASER UNIVERSITY
Fall 2012
All rights reserved.
However, in accordance with the Copyright Act of Canada, this work may be
reproduced without authorization under the conditions for “Fair Dealing.”
Therefore, limited reproduction of this work for the purposes of private study,
research, criticism, review and news reporting is likely to be in accordance
with the law, particularly if cited appropriately.
APPROVAL
Name: Lino Demasi
Degree: Doctor of Philosophy
Title of Thesis: Rooted Minors and Delta-Wye Transformations
Examining Committee: Dr. Nilima Nigam, Associate Professor
Chair
Dr. Bojan Mohar, Senior Supervisor
Professor
Dr. Matt DeVos, Supervisor
Assistant Professor
Dr. Karen Yeats, SFU Examiner
Assistant Professor
Dr. Frederic Havet, External Examiner,
CR1, CNRS, Projet Mascotte, I3S, Sophia-Antipolis
Date Approved: October 2, 2012
ii
Partial Copyright Licence
Abstract
In this thesis, we study terminal minors and delta-wye reducibility. The concept of terminal
minors extends the notion of graph minors to the case where we have a distinguished set
of vertices T in our graph G that must correspond to a distinguished set of vertices Y
in the minor. Delta-wye reducibility concerns the study of how graphs can be reduced
under a set of six operations: the four series-parallel reductions, delta-wye, and wye-delta
transformations.
For terminal minors, we completely characterize when, given a planar graph with four
terminals, we can find a minor of K2,4 in that graph with the four terminal vertices forming
the larger part of the bipartition. This is an extension of a result due to Robertson and
Seymour for the case when a graph contains three terminals. For delta-wye reducibility, we
study the problem of reducibility for the class of graphs consisting of four-terminal planar
graphs. Using the results about rooted K2,4 minors, we are able to characterize when
3-connected graphs in this class are reducible.
iii
Acknowledgements
I would like to first thank my supervisors Bojan Mohar and Matt DeVos for helping me
learn and grow as a mathematician. I would also like to thank the SFU math community
for being a great place to work and study. A special thank you to my wife Kaitlyn for
supporting me as I pursued my dreams. Last, but not least, I would like to thank all my
friends and colleagues who have made my time here memorable.
iv
Contents
Approval ii
Abstract iii
Acknowledgements iv
Contents v
List of Figures vii
1 Preliminaries 1
1.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Rooted Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Disjoint Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Rooted K2,4 Minors in 4-Terminal Planar Graphs 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Low Connectivity Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Disconnected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 1-separations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 2-separations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Three-Connected Graphs and the Main Theorem . . . . . . . . . . . . . . . . 19
2.3.1 Important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
v
2.3.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Proof of the Theorem 2.3.5 . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Algorithm For Finding a Rooted K2,4 Minor . . . . . . . . . . . . . . . . . . . 47
3 Delta-Wye Transformations 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Four Terminal Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Irreducible graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 Cubic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Planar Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Conclusion and Future Work 66
4.1 Rooted K2,4 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Delta Wye Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Appendix A Graph Reductions 69
A.1 Main Theorem Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.1.1 Graphs with OWO Structure . . . . . . . . . . . . . . . . . . . . . . . 70
A.1.2 Graphs with DF Structure . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.1.3 HF Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A.1.4 Graphs with DCJ Structure . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2 Doublecross Graphs and Starfish . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.2.1 Graphs with Doublecross Structure . . . . . . . . . . . . . . . . . . . . 80
A.2.2 Starfish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bibliography 87
vi
List of Figures
2.1 K2,4 obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Low Connectivity Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Vertices with multiple degree 2 neighbours . . . . . . . . . . . . . . . . . . . . 14
2.4 Structure of G when v ∈ S1, w ∈ S2 . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Subcase of G when v ∈ S1, w ∈ S2 . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 (R3) Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 K2,4 obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Possible structures for H′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.9 Graphs where A does not have the K2,2 minor . . . . . . . . . . . . . . . . . 32
2.10 Interesting cases for B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.11 Structures of the 4-cut from an (R2) edge . . . . . . . . . . . . . . . . . . . . 33
2.12 Structures of the 4-cut from an (R1) edge . . . . . . . . . . . . . . . . . . . . 38
2.13 Special case of the 4-cut from an (R1) edge . . . . . . . . . . . . . . . . . . . 40
2.14 J1OWO cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.15 J1HF cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.16 J1DCJ cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.17 J2 Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.18 Cases for a J2 edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1 k × k half grid for k = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 k × k quarter grid for k = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 4 terminal reduction to K4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Reduciton of an extended K4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 4-terminals on a common face . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Construction of a regular tetrahedron . . . . . . . . . . . . . . . . . . . . . . 60
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3.7 Starfish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 Irreducible graph with 2 terminal vertices and 2 terminal faces . . . . . . . . 65
A.1 D2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.2 D3 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.3 D4 Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.4 Diagonal fixing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.5 OWO after reducing to half and quarter grid . . . . . . . . . . . . . . . . . . 71
A.6 OWO Reductions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.7 OWO Reductions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.8 DF afer reducing to quarter grids . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.9 DF Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.10 HF afer reducing to quarter grids and K4 . . . . . . . . . . . . . . . . . . . . 76
A.11 HF Reductions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.12 HF Reductions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.13 DCJ afer reducing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.14 DCJ Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.15 Doublecross reduced to a quarter grid . . . . . . . . . . . . . . . . . . . . . . 80
A.16 Doublecross Reductions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.17 Doublecross Reductions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.18 Starfish Reductions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.19 Starfish Reductions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.20 Starfish Reductions 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
viii
Chapter 1
Preliminaries
1.1 Definitions and Notation
In this section we provide the main definitions and terminology used in the thesis. We use
standard terminology consistent with [17] unless otherwise noted.
We start with a few definitions. A graph G = (V,E) consists of a set V of vertices and
a set E of edges, where each edge consists of two vertices called its endpoints. We use the
notation uv for an edge joining vertices u and v. When such an edge uv exists, the vertices
u and v are said to be adjacent and are incident with the edge uv. A loop is an edge vv ∈ E
from a vertex v to itself. Multiple edges or parallel edges are edges having the same pair of
endpoints. A graph is simple if it has no loops or parallel edges. Graphs in this thesis are
assumed to be simple, except for those in Chapter 3, or where otherwise noted. The degree
of a vertex v, denoted deg(v), is the number of edges incident with v, with loops counted
twice. A subgraph of G is a graph H such that V (H) ⊆ V (G) and E(H) ⊂ E(H). We
denote this by H ⊆ G.
A path is a sequence of distinct vertices with each consecutive pair joined by an edge.
The first and last vertices in the sequence are the endpoints of the path. A cycle is a path
together with an edge between the endpoints. Two paths are internally disjoint if neither
contains a non-endpoint vertex of the other. A graph G is connected if a path exists between
each pair of vertices of G. A component of a graph is a maximal connected subgraph.
For e = uv ∈ E(G), deletion of e is the operation of removing the edge e from E(G).
This is denoted G − e or G\e. Contraction of the edge uv is an operation that replaces
the vertices u and v by a single vertex incident with each edge that was previously incident
1