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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 ..."
Abstract

Cited by 90 (14 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)
, 904
"... Abstract. 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 i ..."
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Abstract. 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.