Clique-width for hereditary graph classes

Dabrowski, Konrad K.; Johnson, Matthew; Paulusma, Daniël

doi:10.1017/9781108649094.002

Cited by 28 publications

(49 citation statements)

References 112 publications

(236 reference statements)

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“…Dichotomy studies for many other problems in mathematics and computer science are available in the literature in general, as well as within the field of graph theory. This includes dichotomy studies related to boundedness of the clique-width [19], boundedness of the chromatic number of digraphs [4], price of connectivity and independence [18,26], and computational complexity of a number of algorithmic problems such as Graph Homomorphism [28], Graph Isomorphism [47], Dominating Set [36], and various coloring [24] and packing problems [14,35].…”

Section: A Dichotomy Resultsmentioning

confidence: 99%

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

Milanič

Pivač

2021

Electron. J. Combin.

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A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.

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Section: A Dichotomy Resultsmentioning

confidence: 99%

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

Milanič

Pivač

2021

Electron. J. Combin.

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“…We prove this by modifying existing graph constructions for proving unbounded clique-width of the whole class (proofs omitted due to space restrictions). Combining these constructions with Theorem 2 and the state-of-art theorem on clique-width from [26] yields the following summary. Theorem 3.…”

Section: P2mentioning

confidence: 89%

“…Graph width parameters [26,39,41,45,54] make such results possible. A graph class has bounded width if there is a constant c such that the width of all its members is at most c. There are several meta-theorems that provide sufficient conditions for a problem to be tractable on a graph class of bounded width.…”

Section: Introductionmentioning

confidence: 99%

“…Monogenic and bigenic graph classes already have a rich structure, and studying their properties has led to deep insights into the complexity of bounding graph parameters and solving graph problems; see e.g. [18,26,37,40] for extensive algorithmic and structural studies and surveys.…”

Section: Introductionmentioning

confidence: 99%

“…[31]) that a monogenic class of graphs has bounded clique-width if and only if it is a subclass of the class G with F G = {P 4 }. The survey [26] gives a state-of-the-art theorem on the boundedness and unboundedness of clique-width of bigenic graph classes. Unlike treewidth, for which a complete dichotomy is known [5], and mim-width, for which there is an infinite number of open cases [15], this state-of-the-art theorem shows that there are still five open cases (up to an equivalence relation).…”

Section: Introductionmentioning

confidence: 99%

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Clique-Width: Harnessing the Power of Atoms

Dabrowski

Masařík

Novotná

et al. 2020

Graph-Theoretic Concepts in Computer Science

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Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class G if they are so on the atoms (graphs with no clique cut-set) of G. Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is (H1, H2)-free if it is both H1-free and H2-free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for (H1, H2)-free graphs, as evidenced by one known example. We prove the existence of another such pair (H1, H2) and classify the boundedness of clique-width on (H1, H2)-free atoms for all but 18 cases.

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Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

Boros

Gurvich

Milanič

2019

Journal of Graph Theory

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A hypergraph is said to be 1-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of 1-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of 1-Spernerness, thresholdness, and 2-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of 1-Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems are based on certain matrix partitions of the corresponding graphs, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems. IntroductionHypergraphs are one of the most fundamental and general combinatorial objects, encompassing various important structures such as graphs, matroids, and combinatorial designs. Results for more specific discrete structures (e.g., graphs) can often be proved more generally, in the context of suitable classes of hypergraphs (see, e.g., Schrijver [57]). Of course, applications of hypergraph theory to structural and optimization aspects of other discrete structures are not restricted to the above phenomenon. Since it is impossible to survey here all kinds of applications of hypergraphs, let us mention only a few more. First, recent work of Hujdurović et al. [37] made use of connections between hypergraphs and binary matrices, acyclic digraphs, and partially ordered sets to study two combinatorial optimization problems motivated by computational biology. Second, a common approach of applying hypergraph theory to graphs is to define and study a hypergraph derived in an appropriate way from a given graph, depending on what type of property of a graph or its vertex or edge subsets one is interested in. This includes matching hypergraphs [1,63], various clique [32,41,47,51,63], independent set [32,63], neighborhood [12,27], separator [13,60], and dominating set hypergraphs [12,13], etc. Close interrelations between hypergraphs and monotone Boolean functions can be useful in such studies, 1 arXiv:1805.03405v3 [math.CO] 29 May 2018 allowing for the transfer and applications of results from the theory of Boolean functions (see, e.g., [21]).In this work, we present several new applications of hypergraphs to graphs. Our starting point is [9] where the class of 1-Sperner hypergraphs was studied. It was shown that such hypergraphs can be decomposed in a particular way and that they form a subclass of the class of threshold hypergraphs studied by Golumbic [31], by Reiterman et al. [56] and, in the equivalent context of threshold monotone Boolean functions, also by Muroga [48] and by Peled and Simeone [52]. (Precise definitions of all relevant concepts will be given in Section 2.)While threshold h...

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Clique-width for hereditary graph classes

Cited by 28 publications

References 112 publications

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

Clique-Width: Harnessing the Power of Atoms

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

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