Unavoidable induced subgraphs in large graphs with no homogeneous sets

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from $K_{1,n}$ by subdividing every edge once, (2) the line graph of $K_{2,n}$, (3) the line graph of the graph in (1), (4) the half-graph of height $n$, (5) a prime graph induced by a chain of length $n$, (6) two particular graphs obtained from the half-graph of height $n$ by making one side a clique and adding one vertex.Comment: 13 pages, 3 figure

Extremal Problems in Matroid Connectivity

Matroid k-connectivity is typically defined in terms of a connectivity function. We can also say that a matroid is 2-connected if and only if for each pair of elements, there is a circuit containing both elements. Equivalently, a matroid is 2-connected if and only if each pair of elements is in a certain 2-element minor that is 2-connected. Similar results for higher connectivity had not been known. We determine a characterization of 3-connectivity that is based on the containment of small subsets in 3-connected minors from a given list of 3-connected matroids. Bixby’s Lemma is a well-known inductive tool in matroid theory that says that each element in a 3-connected matroid can be deleted or contracted to obtain a matroid that is 3-connected up to minimal 2-separations. We consider the binary matroids for which there is no element whose deletion and contraction are both 3-connected up to minimal 2-separations. In particular, we give a decomposition for such matroids to establish that any matroid of this type can be built from sequential matroids and matroids with many fans using a few natural operations. Wagner defined biconnectivity to translate connectivity in a bicircular matroid to certain connectivity conditions in its underlying graph. We extend a characterization of biconnectivity to higher connectivity. Using these graphic connectivity conditions, we call upon unavoidable minor results for graphs to find unavoidable minors for large 4-connected bicircular matroids

Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

Suppose F is a finite family of graphs. We consider the following meta-problem, called F-Immersion Deletion: given a graph G and an integer k, decide whether the deletion of at most k edges of G can result in a graph that does not contain any graph from F as an immersion. This problem is a close relative of the F-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from F. We prove that whenever all graphs from F are connected and at least one graph of F is planar and subcubic, then the F-Immersion Deletion problem admits: - a constant-factor approximation algorithm running in time O(m^3 n^3 log m) - a linear kernel that can be computed in time O(m^4 n^3 log m) and - a O(2^{O(k)} + m^4 n^3 log m)-time fixed-parameter algorithm, where n,m count the vertices and edges of the input graph. Our findings mirror those of Fomin et al. [FOCS 2012], who obtained similar results for F-Minor Deletion, under the assumption that at least one graph from F is planar. An important difference is that we are able to obtain a linear kernel for F-Immersion Deletion, while the exponent of the kernel of Fomin et al. depends heavily on the family F. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of F-Immersion Deletion is quite different than that of F-Minor Deletion

Unavoidable Minors of Graphs of Large Type.

In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is defined to be the minimum number n such that there is a sequence of graphs G = G\sb0, G\sb1,\... , G\sb{n}, where G\sb{i} is obtained by contracting or deleting one edge from each block of G\sb{i-1}, and where G\sb{n} is edgeless. We show that a 3-connected graph has large type if and only if it has a minor isomorphic to a large fan. Furthermore, we show that if a graph has large type, then it has a minor isomorphic to a large fan or to a large member of one of two specified families of graphs

Capturing elements in matroid minors

In this dissertation, we begin with an introduction to a matroid as the natural generalization of independence arising in three different fields of mathematics. In the first chapter, we develop graph theory and matroid theory terminology necessary to the topic of this dissertation. In Chapter 2 and Chapter 3, we prove two main results. A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n exceeding two, there is an integer f(n) so that if |E(M)| exceeds f(n), then M has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{3,n}, or U_{2,n} or U_{n-2,n}. In Chapter 2, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if |E(M)| exceeds g(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{1,1,1,n}, a specific single-element extension of M(K_{3,n}) or the dual of this extension, or U_{2,n} or U_{n-2,n}. In Chapter 3, we consider a large 3-connected binary matroid with a specified pair of elements. We extend a corollary of the result of Chapter 2 to show the following result for any pair {x,y} of elements of a 3-connected binary matroid M. For every integer n exceeding two, there is an integer h(n) so that if |E(M)| exceeds h(n), then x and y are elements of a minor of M isomorphic to the rank-n wheel, a rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K_{1,1,1,n}

Obstructions for bounded shrub-depth and rank-depth

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hlin\v{e}n\'y, Kwon, Obdr\v{z}\'alek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.Comment: 19 pages, 5 figures; accepted to Journal of Combinatorial Theory Ser.