On extended P4-reducible and extended P4-sparse graphs

Giakoumakis, Vassilis; Vanherpe, Jean-Marie

doi:10.1016/s0304-3975(96)00220-4

Cited by 36 publications

(21 citation statements)

References 11 publications

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“…Thus, G(t) is isomorphic to the graph induced by a subset of M (t) consisting of a single vertex from each maximal strong submodule of M (t) in the modular decomposition of G. Depending on whether an internal node t of T (G) is a P-, S-, or N-node, the following result holds (see also [10]):…”

Section: Modular Decomposition and P 4 -Sparse Graphsmentioning

confidence: 99%

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A Fully Dynamic Algorithm for the Recognition of P 4-Sparse Graphs

Nikolopoulos

Palios

Papadopoulos

2006

Graph-Theoretic Concepts in Computer Science

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Abstract:In this paper, we solve the dynamic recognition problem for the class of P 4 -sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P 4 -sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish properties for each modification to yield a P 4 -sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for P 4 -sparse graphs.

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Section: Modular Decomposition and P 4 -Sparse Graphsmentioning

confidence: 99%

“…In particular, for the class of P 4 -sparse graphs, Giakoumakis and Vanherpe [10] showed that: Lemma 2.2. Let G be a graph and let T (G) be its modular decomposition tree.…”

Section: Modular Decomposition and P 4 -Sparse Graphsmentioning

confidence: 99%

A Fully Dynamic Algorithm for the Recognition of P 4-Sparse Graphs

Nikolopoulos

Palios

Papadopoulos

2006

Graph-Theoretic Concepts in Computer Science

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“…Thus, G(t) is isomorphic to the graph induced by a subset of M (t) consisting of a single vertex from each maximal strong submodule of M (t) in the modular decomposition of G. For the P-, S-, and N-nodes, the following lemma holds (see also [8]):…”

Section: Modular Decompositionmentioning

confidence: 99%

“…Regarding the modular decomposition of P 4 -sparse graphs, Giakoumakis and Vanherpe [8] showed the following result (recall that the graph G(t) has vertices the children of the node t in T (G)):…”

Section: Lemma 22 (Theorem 1 In [12]) For a Graph G The Following mentioning

confidence: 99%

“…Moreover, in [12] the same authors provided efficient solutions to other classical optimization problems; that is, finding the clique number, the stable number, the chromatic number and the clique cover number of a P 4 -sparse graph. Giakoumakis and Vanherpe [8] obtained lineartime algorithms for the maximum weight clique and for the maximum weight stable set problems on P 4 -sparse graphs using the modular decomposition tree representation [5,15]. Yang and Yu [18] exhibited a linear time algorithm for the maximum matching problem in cographs, while Fouquet, Parfenoff, and Thuillier [7] extended this algorithm to P 4 -tidy graphs, a class containing both P 4 -sparse graphs and cographs.…”

Section: Introductionmentioning

confidence: 99%

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Maximum-Size Subgraphs of P4-Sparse Graphs Admitting a Perfect Matching

Nikolopoulos

Palios

2005

Advances in Informatics

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Abstract. In this paper, we address the problem of computing a maximum-size subgraph of a P4-sparse graph which admits a perfect matching; in the case where the graph has a perfect matching, the solution to the problem is the entire graph. We establish a characterization of such subgraphs, and describe an algorithm for the problem which for a P4-sparse graph on n vertices and m edges, runs in O(n + m) time and space. The above results also hold for the class of complement reducible graphs or cographs, a well-known subclass of P4-sparse graphs.

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Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

Courcelle

Makowsky

Rotics

1998

Graph-Theoretic Concepts in Computer Science

245

383

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Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of tree-width at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic second-order logic has a linear algorithm. We prove that this is also the case for graphs of clique-width at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with "too many" induced paths with four vertices.

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On extended P4-reducible and extended P4-sparse graphs

Cited by 36 publications

References 11 publications

A Fully Dynamic Algorithm for the Recognition of P 4-Sparse Graphs

A Fully Dynamic Algorithm for the Recognition of P 4-Sparse Graphs

Maximum-Size Subgraphs of P4-Sparse Graphs Admitting a Perfect Matching

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

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