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Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 111 (12 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
SEIDEL MINOR, PERMUTATION GRAPHS AND COMBINATORIAL PROPERTIES (EXTENDED ABSTRACT)
, 2009
"... A permutation graph is an intersection graph of segments lying between two parallel lines. A Seidel complementation of a finite graph at one of it vertex v consists to complement the edges between the neighborhood and the nonneighborhood of v. Two graphs are Seidel complement equivalent if one can ..."
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A permutation graph is an intersection graph of segments lying between two parallel lines. A Seidel complementation of a finite graph at one of it vertex v consists to complement the edges between the neighborhood and the nonneighborhood of v. Two graphs are Seidel complement equivalent if one can be obtained from the other by a successive application of Seidel complementation. In this paper we introduce the new concept of Seidel complementation and Seidel minor, we then show that this operation preserves cographs and the structure of modular decomposition. The main contribution of this paper is to provide a new and succinct characterization of permutation graphs i.e. A graph is a permutation graph if and only if it does not contain the following graphs: C5, C7, XF2 6, XF 2n+3 5, C2n, n � 6 and their complement as Seidel minor. In addition we provide a O(n + m)time algorithm to output one of the forbidden Seidel minor if the graph is not a permutation graph.
On classes of graphs with logarithmic booleanwidth
"... Booleanwidth is a recently introduced graph parameter. Many problems are fixed parameter tractable when parametrized by booleanwidth, for instance "Minimum Weighted Dominating Set " (MWDS) problem can be solved in O∗(23k) time given a booleandecomposition of width k, hence for all gra ..."
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Booleanwidth is a recently introduced graph parameter. Many problems are fixed parameter tractable when parametrized by booleanwidth, for instance "Minimum Weighted Dominating Set " (MWDS) problem can be solved in O∗(23k) time given a booleandecomposition of width k, hence for all graph classes where a booleandecomposition of width O(log n) can be found in polynomial time, MWDS can be solved in polynomial time. We study graph classes having booleanwidth O(log n) and problems solvable in O∗(2O(k)), combining these two results to design polynomial algorithms. We show that for trapezoid graphs, circular permutation graphs, convex graphs, Dilworthk graphs, circular arc graphs and complements of kdegenerate graphs, booleandecompositions of width O(log n) can be found in polynomial time. We also show that circular ktrapezoid graphs have booleanwidth O(log n), and find such a decomposition if a circular ktrapezoid intersection model is given. For many of the graph classes we also prove that they contain graphs of booleanwidth Θ(log n). Further we apply the results from [1] to give a new polynomial time algorithm solving all vertex partitioning problems introduced by Proskurowski and Telle [23]. This extends previous results by Kratochvíl, Manuel and Miller [14] showing that a large subset of the vertex partitioning problems are polynomial solvable on interval graphs. 1