Showing papers on "K-tree published in 1988"
TL;DR: The notion of the boundary clique and the k-overlap clique graph are introduced and the following are proved: Every incomplete chordal graph has two nonadjacent simplicial vertices lying in boundary cliques.
Abstract: We introduce the notion of the boundary clique and the k-overlap clique graph and prove the following: Every incomplete chordal graph has two nonadjacent simplicial vertices lying in boundary cliques. An incomplete chordal graph G is k-connected if and only if the k-overlap clique graph gk(G) is connected. We give an algorithm to construct a clique tree of a connected chordal graph and characterize clique trees of connected chordal graphs using the algorithm.
64 citations
TL;DR: An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G.
33 citations
TL;DR: This paper proposes another parallel algorithm for maximal cliques which can be executed in O(log2n) time by using only O(n3) processors and proposes another two algorithms for computing a clique tree and minimum coloring which are more efficient than those proposed by Naor et al.
25 citations
TL;DR: If in a clique on n vertices, the edges between cn a vertices are deleted,1/2⩽a≤1, then the number of cliques needed to partition what is left is asymptotic to c 2 n 2a ; this fills in a gap between results of Wallis and Pullman and Donald for a=1, c=1.
23 citations
[...]
Abstract: Let P be an undirected graph with vertices V and edges E. Fix an enumeration, {v1,v2,...,vn}, of V and let M(P) = {A ∈ Mn (ℂ)| = 0 if (vi,vj) ∉ E where ei is the standard orthonormal basis of ℂn. Mn (ℂ)+ is the set of positive semi-definite n × n matrices with complex entries. For X ⊂ Mn (ℂ)+ a cone, define the order of X, denoted ord(X), to be the smallest integer k such that the elements of X of rank at most k generate X as a cone. For any set X, let Mm (X) denote m x m matrices with entries from X. It is known that a graph P is chordal if and only if ord(Mm (M(P))+) = 1 for every positive integer m, where Mm (M(P))+ = {A ∈ Mm (M(P))|A is positive semi-definite}. We characterize, in a graph theoretic way, graphs P for which ord(Mm (M(P))+) = ord(M(P)+) ≤ 2 for every positive integer m.
7 citations
TL;DR: If G is chordal and has maximum clique size ω(G) = m, then i (G) ⩽ [1 + o(1)]m/log2m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).
Abstract: The interval number of a (simple, undirected) graph G is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t real intervals. A chordal (or triangulated) graph is one with no induced cycles on 4 or more vertices. If G is chordal and has maximum clique size ω(G) = m, then i(G) ⩽ [1 + o(1)]m/log2m and this result is best possible, even for split graphs (chordal graphs whose complement is also chordal).
6 citations
21 Dec 1988
TL;DR: The two path problem (TPP) is to determine whether there exist two vertex disjoint paths connecting s with t and u with v and to find such paths if they exist.
Abstract: Let G= (V,E) be a finite undirected graph with four distinguished vertices s, t, u, v. The two path problem (TPP) is to determine whether there exist two vertex disjoint paths connecting s with t and u with v and to find such paths if they exist.
4 citations