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40
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 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.
Interval bigraphs and circular arc graphs
 J. Graph Theory
"... Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hop ..."
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Cited by 19 (5 self)
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Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidaltriplefree graphs, permutation graphs, and cocomparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of M"uller. 1 Background A graph H is an interval graph if it is the intersection graph of a family of intervals Iv, v 2 V (H). (Two vertices v; v 0 are adjacent in H if and only if Iv and Iv0 intersect.) If the
A simple test for interval graphs
 Proc. 18th Int. Workshop (WG '92), GraphTheoretic Concepts in Computer Science
, 1992
"... An interval graph is the intersection graph of a collection of intervals. Interval graphs are a special class of chordal graphs. This class of graphs has a wide range of applications. Several linear time algorithms have been designed to recognize interval graphs. Booth & Lueker first used PQtrees t ..."
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Cited by 19 (2 self)
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An interval graph is the intersection graph of a collection of intervals. Interval graphs are a special class of chordal graphs. This class of graphs has a wide range of applications. Several linear time algorithms have been designed to recognize interval graphs. Booth & Lueker first used PQtrees to recognize interval graphs in linear time. However, the data manipulation of PQtrees is rather involved and the complexity analysis is also quite tricky. Korte and Möhring simplified the operations on a PQtree using an incremental algorithm. Hsu and Ma gave a simpler decomposition algorithm without using PQtrees. All of these algorithms rely on the following fact: a graph is an interval graph iff there exists a linear order of its maximal cliques such that for each vertex v, all maximal cliques containing v are consecutive. Thus, the precomputation of all maximal cliques is required for these algorithms. Based on graph decomposition, we give a much simpler recognition algorithm in this paper which directly places the intervals without precomputing all maximal cliques. A linear time isomorphism algorithm can be easily derived as a byproduct. Another advantage of our approach is that it can be used to develop an O(nlog n) online recognition algorithm for interval graphs. 1.
Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs
 SIAM Journal on Computing
, 1999
"... Abstract. In this paper, we present a lineartime algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a lineartime algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O( ..."
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Cited by 16 (3 self)
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Abstract. In this paper, we present a lineartime algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a lineartime algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O(n 2)toO(n+m). We also devise a simple lineartime algorithm for interval graph recognition where no complicated data structure is involved. Key words. chordal graph, triangulated graph, interval graph, analysis of algorithms, graph theory, substitution decomposition, modular decomposition, cyclefree poset, transitive orientation, graph partitioning, cardinality lexicographic ordering, graph recognition
A dichotomy for minimum cost graph homomorphisms
 European J. Combin
, 2007
"... For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost h ..."
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Cited by 14 (6 self)
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For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs ci(u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NPhard. This solves an open problem from an earlier paper. 1
Optimal reduction of twoterminal directed acyclic graphs
 SIAM Journal on Computing
, 1992
"... Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O ..."
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Cited by 14 (1 self)
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Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2"5) algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NPhard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly seriesparallel.
Algorithmic Combinatorics based on Slicing Posets
, 2002
"... We show that some recent results in slicing of a distributed computation can be applied to developing algorithms to solve problems in combinatorics. A combinatorial problem usually requires enumerating, counting or ascertaining existence of structures that satisfy a given property B. We cast the com ..."
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Cited by 7 (6 self)
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We show that some recent results in slicing of a distributed computation can be applied to developing algorithms to solve problems in combinatorics. A combinatorial problem usually requires enumerating, counting or ascertaining existence of structures that satisfy a given property B. We cast the combinatorial problem as a distributed computation such that there is a bijection between combinatorial structures satisfying B and the global states that satisfy a property equivalent to B. We then apply results in slicing a computation with respect to a predicate to obtain a small representation of only those global states that satisfy B.
Linear Structure of Bipartite Permutation Graphs and the Longest Path Problem
"... The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation gr ..."
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Cited by 7 (0 self)
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The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph.