Claw-free graphs. V. Global structure

Chudnovsky, Maria; Seymour, Paul

doi:10.1016/j.jctb.2008.03.002

Cited by 81 publications

(187 citation statements)

References 4 publications

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“…Chudnovsky and Seymour, in a series of papers [30,31,32,33,34,35,36], extend the above structural characterization of claw-free Berge graphs, to claw-free graphs in general. They show how all claw-free graphs can be obtained through explicit constructions starting from a few basic classes that can all be described explicitly.…”

Section: Corollary 85 Ifmentioning

confidence: 88%

“…The full structural characterization they obtain is too complicated to explain. Here we state the decomposition theorem they obtain in [33] and then use in [34] for describing the construction. Basic claw-free graphs consist of seven subclasses, some of which are line graphs (of multigraphs), induced subgraphs of icosahedron, circular interval graphs and antiprismatic graphs (claw-free graphs in which every four vertices induce a subgraph with at least two edges).…”

Section: Corollary 85 Ifmentioning

confidence: 99%

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The world of hereditary graph classes viewed through Truemper configurations

Vušković¹

2013

Surveys in Combinatorics 2013

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In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few specific types, that we will call Truemper configurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices).The configurations that Truemper identified in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper configurations, focusing on the algorithmic consequences, trying to understand what structurally enables efficient recognition and optimization algorithms.

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Section: Corollary 85 Ifmentioning

confidence: 88%

Section: Corollary 85 Ifmentioning

confidence: 99%

The world of hereditary graph classes viewed through Truemper configurations

Vušković¹

2013

Surveys in Combinatorics 2013

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“…In an obvious way, our algorithm can be used to find a minimum weight vertex cover in G or a maximum weight clique in the complement of G. We also believe that the structural analysis given in this paper can be used to extend many of the combinatorial properties of claw-free graphs established in the recent line of research by Chudnovsky and Seymour [8,7,9,10,11,12,13,14] to apple-free graphs.…”

mentioning

confidence: 78%

Independent Sets of Maximum Weight in Apple-Free Graphs

Brandstädt

Klembt

Mosca

2008

Algorithms and Computation

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Abstract. We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free.

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“…In this problem, by applying the same vertex modulator principle we arrive at the situation where we have a modulator X ⊆ V (G) with |X| ≤ 4k, and G − X is a claw-free graph. Then, one can use the structural theorem of Chudnovsky and Seymour [8,9] (see also variants suited for algorithmic applications, e.g., [21]) to understand the structure of G − X and of the adjacencies between X and G − X. In essence, the structural theorem yields a decomposition of G−X into strips, where each strip induces a graph from one of several basic graph classes; each strip has at most two distinguished cliques (possibly equal) called ends, and strips are joined together by creating full…”

Section: (G)mentioning

confidence: 99%

Polynomial Kernelization for Removing Induced Claws and Diamonds

Cygan

Pilipczuk

et al. 2016

Graph-Theoretic Concepts in Computer Science

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A graph is called {claw, diamond}-free if it contains neither a claw (a K 1,3 ) nor a diamond (a K 4 with an edge removed) as an induced subgraph. Equivalently, {claw, diamond}-free graphs are characterized as line graphs of triangle-free graphs, or as linear dominoes (graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique). We consider the parameterized complexity of the {CLAW,DIAMOND}-FREE EDGE DELETION problem, where given a graph G and a parameter k, the question is whether one can remove at most k edges from G to obtain a {claw, diamond}-free graph. Our main result is that this problem admits a polynomial kernel. We complement this result by proving that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time 2 o(k) · |V (G)| O(1) unless the Exponential Time Hypothesis fails.

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Claw-free graphs. V. Global structure

Cited by 81 publications

References 4 publications

The world of hereditary graph classes viewed through Truemper configurations

The world of hereditary graph classes viewed through Truemper configurations

Independent Sets of Maximum Weight in Apple-Free Graphs

Polynomial Kernelization for Removing Induced Claws and Diamonds

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