Showing papers on "K-tree published in 2002"
TL;DR: It is shown that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs.
Abstract: We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time Our approach unifies these algorithms Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs for which the treewidth and the minimum fill-in problems were open
214 citations
TL;DR: In this article, it was shown that the potential maximal cliques of a graph can be generated in polynomial time in the number of maximal separators of the graph, and that the treewidth and the minimum fill-in are polynomially tractable for all classes of graphs with a constant number of minimal separators.
114 citations
TL;DR: In this article, the bicolorability of a clique hypergraph C(G) is studied and the question of whether the vertices of G can be colored with two colors so that no maximal clique is monochromatic is solved in polynomial time.
71 citations
TL;DR: It is proved that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect, which leads to polynomial time algorithms for finding the parameters τC (G) and αC(G), for graphs belonging to this class.
Abstract: A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A clique-independent set is a subset of pairwise disjoint cliques of G. Denote by τ
C
(G) and α
C
(G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τ
C
(H)=α
C
(H), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters τ
C
(G) and α
C
(G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference τ
C
(G)−α
C
(G) is arbitrarily large.
39 citations
TL;DR: It is shown that all clique trees of a chordal graph can be obtained from the reduced clique hypergraph; thus the reduced adjacency hypergraph can be thought of as a generalization of the notion of a clique tree.
28 citations
TL;DR: An important family of graphs is introduced which is closed under the clique operator and contains clique divergent graphs with strictly linear growth, i.e., o(knG) = o(G) + rn, where r is any fixed positive integer.
27 citations
TL;DR: To find all maximal cliques of a trapezoid graph a set of intervals have been constructed by projecting the geometrical representation of the graph on the bottom line by taking O(n 2 + yn) time.
Abstract: In this paper, to find all maximal cliques of a trapezoid graph a set of intervals have been constructed by projecting the geometrical representation of the graph on the bottom line. The proposed algorithm for this purpose takes O(n 2 + yn) time, where n is the number of vertices of the graph and y is the output size.
18 citations
TL;DR: Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a k -tree, a subclass of the class of chordal graphs, for the fast reordering problem and the isomorphism problem.
Abstract: In this paper two problems on the class of k -trees, a subclass of the class of chordal graphs, are considered: the fast reordering problem and the isomorphism problem. An O(log 2n) time parallel algorithm for the fast reordering problem is described that uses O(nk(n-k)/\kern -1ptlog n) processors on a CRCW PRAM proving membership in the class NC for fixed k . An O(nk(k+1)!) time sequential algorithm for the isomorphism problem is obtained representing an improvement over the O(n2k(k+1)!) algorithm of Sekharan (the second author) [10]. A parallel version of this sequential algorithm is presented that runs in O(log 2n) time using O((nk((k+1)!+n-k))/log n) processors improving on a parallel algorithm of Sekharan for the isomorphism problem [10]. Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a k -tree.
11 citations
TL;DR: It is shown that a known necessary condition for a graph to be an edge clique graph is that the sizes of all maximal cliques and intersections of maximal clique ought to be triangular numbers, and that this condition is also sufficient for starlike-threshold graphs.
10 citations
TL;DR: An optimal parallel algorithm is proposed on the EREW PRAM for testing isomorphism of two maximal outerplanar graphs and it is shown that it takes O(logn) time using O(n) work.