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 = G0, G1,...,Gn, where Gi is obtained by contracting one edge in or deleting one edge from each block of G/_i, and where G 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. © 2002 Elsevier Science B.V. All rights reserved

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

Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure

First order convergence of matroids

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini-Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed.Comment: Accepted to the European Journal of Combinatoric

Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio

A characterization of almost all minimal not nearly planar graphs

In this dissertation, we study nearly planar graphs, that is, graphs that are edgeless or have an edge whose deletion results in a planar graph. We show that all but finitely many graphs that are not nearly planar and do not contain one particular graph have a well-understood structure based on large Möbius ladders

Book embeddings of graphs

We use a structural theorem of Robertson and Seymour to show that for every minor-closed class of graphs, other than the class of all graphs, there is a number k such that every member of the class can be embedded in a book with k pages. Book embeddings of graphs with relation to surfaces, vertex extensions, clique-sums and r-rings are combined into a single book embedding of a graph in the minor-closed class. The effects of subdividing a complete graph and a complete bipartite graph with respect to book thickness are studied. We prove that if n ≥ 3, then the book thickness of Kn is the ceiling of (n/2). We also prove that for each m and B, there exists an integer N such that for all n ≥ ‪N, the book thickness of the graph obtained from subdividing each edge of Kn exactly m times has book thickness at least B. Additionally, there are corresponding theorems for complete bipartite graphs

Parameterized complexity : permutation patterns, graph arrangements, and matroid parameters

The theory of parameterized complexity is an area of computer science focusing on refined analysis of hard algorithmic problems. In the thesis, we give two complexity lower bounds and define two novel parameters for matroids. The first lower bound is a kernelization lower bound for the Permutation Pattern Matching problem, which is concerned with finding a permutation pattern inside another input permutation. Our result states that unless a certain (widely believed) complexity hypothesis fails, it is impossible to construct a polynomial time algorithm taking an instance of the Permutation Pattern Matching problem and producing an equivalent instance of size bounded by a polynomial of the length of the pattern. Obtaining such lower bounds has been posed by Stephane Vialette as an open problem. We then prove a subexponential lower bound for the computational complexity of the Optimum Linear Arrangement problem. In our theorem, we assume a conjecture about the computational complexity of a variation of the Min Bisection problem. The two matroid parameters introduced in this work are called amalgam-width and branch-depth. Amalgam-width is a generalization of the branch-width parameter that allows for algorithmic applications even for matroids that are not finitely representable. We prove several results, including a theorem stating that deciding monadic second-order properties is fixed-parameter tractable for general matroids parameterized by amalgam-width. Branch-depth, the other newly introduced matroid parameter, is an analogue of graph tree-depth. We prove several statements relating graph tree-depth and matroid branch-depth. We also present an algorithm that efficiently approximates the value of the parameter on a general oracle-given matroid