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

A Characterization of Uniquely Representable Graphs

The betweenness structure of a finite metric space $M = (X, d)$ is a pair $\mathcal{B}(M) = (X,\beta_M)$ where $\beta_M$ is the so-called betweenness relation of $M$ that consists of point triplets $(x, y, z)$ such that $d(x, z) = d(x, y) + d(y, z)$. The underlying graph of a betweenness structure $\mathcal{B} = (X,\beta)$ is the simple graph $G(\mathcal{B}) = (X, E)$ where the edges are pairs of distinct points with no third point between them. A connected graph $G$ is uniquely representable if there exists a unique metric betweenness structure with underlying graph $G$. It was implied by previous works that trees are uniquely representable. In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs. Further, we prove that two related classes of graphs coincide with the class of block graphs and the class of distance-hereditary graphs, respectively. We show that our results hold not only for metric but also for almost-metric betweenness structures.Comment: 16 pages (without references); 3 figures; major changes: simplified proofs, improved notations and namings, short overview of metric graph theor

Linked Tree-Decompositions of Infinite Represented Matroids

It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids. Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width

On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results

Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes. The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results

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

Structure and algorithms for (cap, even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48

The Sandwich Problem for Decompositions and Almost Monotone Properties

This is a post-peer-review, pre-copyedit version of an article published in Algorithmica. The final authenticated version is available online at: https://doi.org/10.1007/s00453-018-0409-6We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in C as an induced subgraph is almost monotone if C is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length ≡ j mod n is almost monotone if and only if j ≡ 2 mod n or n ≤ 2. Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.Maria Chudnovsky was supported by National Science Foundation Grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. Celina M. H. de Figueiredo was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq Grant 303622/2011-3

Critical properties of bipartite permutation graphs

The class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique‐ and rank‐width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique‐ and rank‐width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub‐depth, and with respect to other notions, such as well‐quasi‐ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite permutation graphs of various types