A chain theorem for 3 \u3csup\u3e+\u3c/sup\u3e-connected graphs

A 3-connected graph is called 3 +-connected if it has no 3-separation that separates a large fan or K 3,n from the rest of the graph. It is proved in this paper that except for K 4, every 3 +-connected graph has a 3 +-connected proper minor that is at most two edges away from the original graph. This result is used to characterize Q-minor-free graphs, where Q is obtained from the cube by contracting an edge. © 2012 Society for Industrial and Applied Mathematics

A chain theorem for 4-connected graphs

A sequence of 4-connected graphs G0,G1,.,Gn is called a (G0,Gn)-chain if each Gi (

On Binary And Regular Matroids Without Small Minors

The results of this dissertation consist of excluded-minor results for Binary Matroids and excluded-minor results for Regular Matroids. Structural theorems on the relationship between minors and k-sums of matroids are developed here in order to provide some of these characterizations. Chapter 2 of the dissertation contains excluded-minor results for Binary Matroids. The first main result of this dissertation is a characterization of the internally 4-connected binary matroids with no minor that is isomorphic to the cycle matroid of the prism+e graph. This characterization generalizes results of Mayhew and Royle [18] for binary matroids and results of Dirac [8] and Lovász [15] for graphs. The results of this chapter are then extended from the class of internally 4-connected matroids to the class of 3-connected matroids. Chapter 3 of the dissertation contains the second main result, a decomposition theorem for regular matroids without certain minors. This decomposition theorem is used to obtain excluded-minor results for Regular Matroids. Wagner, Lovász, Oxley, Ding, Liu, and others have characterized many classes of graphs that are H-free for graphs H with at most twelve edges (see [7]). We extend several of these excluded-minor characterizations to regular matroids in Chapter 3. We also provide characterizations of regular matroids excluding several graphic matroids such as the octahedron, cube, and the Möbius Ladder on eight vertices. Both theoretical and computer-aided proofs of the results of Chapters 2 and 3 are provided in this dissertation

Circumference and Pathwidth of Highly Connected Graphs

Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t-1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger result. Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007] who showed that every graph without k disjoint cycles of length at least t has bounded treewidth (as a function of k and t). Our main result states that, under the additional assumption of (k + 1)- connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover, examples show that (k + 1)-connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which values of k and graphs H does every k-connected H-minor-free graph have bounded pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure

Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects

Disjoint Essential Cycles

AbstractGraphs that have two disjoint noncontractible cycles in every possible embedding in surfaces are characterized. Similar characterization is given for the class of graphs whose orientable embeddings (embeddings in surfaces different from the projective plane, respectively) always have two disjoint noncontractible cycles. For graphs which admit embeddings in closed surfaces without having two disjoint noncontractible cycles, such embeddings are structurally characterized