On Structures of Large Rooted Graphs

A rooted graph is a pair (G,R), where G is a graph and R⊆V(G). There are two research topics in this thesis. One is about unavoidable substructures in sufficiently large rooted graphs. The other is about characterizations of rooted graphs excluding specific large graphs. The first topic of this thesis is motivated by Ramsey Theorem, which states that K_n and ¯(K_n ) are unavoidable induced subgraphs in every sufficiently large graph. It is also motivated by a classical result of Oporowski, Oxley, and Thomas, which determines unavoidable large 3-connected minors. We first determine unavoidable induced subgraphs, and unavoidable subgraphs in connected graphs with sufficiently many roots. We also extend this result to generalized rooted connected graphs. Secondly, we extend these results to rooted graphs of higher connectivity. In particular, we determine unavoidable subgraphs of sufficiently large rooted 2- connected graphs. Again, this result is extended to generalized rooted 2-connected graphs. The second topic of this dissertation is motivated by two results of Robertson and Seymour, let’s only talk about path and star. In the first result they established that graphs without a long path subgraph are precisely those that can be constructed using a specific operation within a bounded number of iterations, starting from the trivial graph. In the second result they showed that graphs without a large star minor are those that are subdivisions of graphs with bounded number vertices. We consider similar problems for path, star and comb. We have some theorems on characterizations of rooted connected graphs excluding a heavy path, a large (nicely) confined comb, a large (nicely) confined star, which are similar to those of Robertson and Seymour. Moreover, our results strengthen their related results

Unavoidable minors in graphs and matroids

It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges to be contracted when they meet a degree-2 vertex. Dually, a parallel minor allows for any edge contractions but restricts the deletion of edges to those that lie in 2-edge cycles. This dissertation begins by proving the dual results to those noted above. These identify all unavoidable parallel minors for finite graphs of low connectivity. Following this, corresponding results on unavoidable minors for infinite graphs are proved. The dissertation concludes by finding the unavoidable parallel minors for 3-connected regular matroids, which combines the results for unavoidable series and parallel minors for graphs with Seymour\u27s decomposition theorem for regular matroids

Unavoidable Structures in Large and Infinite Graphs

In this work, we present results on the unavoidable structures in large connected and large 2-connected graphs. For the relation of induced subgraphs, Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either Kr or Kr. It is well known that, for every positive integer r, every sufficiently large connected graph contains an induced subgraph isomorphic to one of Kr, K1,r, and Pr. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. Using some techniques from the finite result, we give the unavoidable induced subgraphs of infinite 2-connected graphs. We then shift our attention to the relation of bipartite minors defined in 2016 by Chudnovsky, Kalai, Nevo, Novik, and Seymour. For the relation of bipartite minors, we present the unavoidable substructures of both large connected and large 2-connected bipartite graphs

Unavoidable topological minors of infinite graphs

This is the post-print version of the Article - Copyright @ 2010 ElsevierA graph G is loosely-c-connected, or ℓ-c-connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of ℓ-c-connected infinite graphs that form an unavoidable set among the topological minors of ℓ-c-connected infinite graphs. Corresponding results for minors and parallel minors are also obtained.This study was supported in part by NSF grants DMS-1001230 and NSA grant H98230-10-1-018

Unavoidable Immersions and Intertwines of Graphs

The topological minor and the minor relations are well-studied binary relations on the class of graphs. A natural weakening of the topological minor relation is an immersion. An immersion of a graph H into a graph G is a map that injects the vertex set of H into the vertex set of G such that edges between vertices of H are represented by pairwise-edge-disjoint paths of G. In this dissertation, we present two results: the first giving a set of unavoidable immersions of large 3-edge-connected graphs and the second on immersion intertwines of infinite graphs. These results, along with the methods used to prove them, are analogues of results on the graph minor relation. A conjecture for the unavoidable immersions of large 3-edge-connected graphs is also stated with a partial proof

Graph-based matching of occluded hand gestures

Occlusion is an unavoidable subject in most machine vision areas. Recognition of partially-occluded hand gestures is an important problem. In this paper a new algorithm is proposed for the recognition of occluded and non-occluded hand gestures based on matching the Graphs of gestures in an eigenspac

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

Let G be a class of graphs with a membership test, k∈N , and let Gk be the class of graphs in G of path-width at most k. We present an interactive framework that finds an unavoidable set for Gk, which is a set of graphs U such that any graph in Gk contains an isomorphic copy of a graph in U. At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for Gk. While obstruction sets are well-studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever G is locally checkable of bounded maximum degree and U is a finite set of connected graphs. For example, l-regular graphs, l-colourable graphs, and H-free graphs are locally checkable classes. We put special emphasis on the case that G is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path-width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path-width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path-width 3 and girth 4