Showing papers on "K-tree published in 1993"
TL;DR: An O( nm )-time algorithm to decompose a graph with n vertices and m edges by means of clique separators is described, modified, so that no new maximal prime subgraphs are generated, i.e. so that a graph is decomposed exactly into its maximal primeSubgraphs which is the unique minimal derived system of prime sub graphs.
130 citations
TL;DR: This work is motivated by a resource allocation problem dealing with a partially replicated distributed database defined on a tree network and gives a one-pass algorithm for finding the core of a tree and a generalization of a core which is a k -tree core.
57 citations
01 Jul 1993
TL;DR: There exist visibility graphs of nonintersecting line segments inthe plane whose smallest clique cover has size, and it is shown that the visibilitygraph of a simple polygon always admits aClique cover of size.
Abstract: We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graph G, a family G={G1,G2,...,Gk} is called a clique cover of G if (i) each Gi is a clique or a bipartite clique, and (ii) the union of Gi is G. The size of the clique cover G is defined as Σki=1 ni, where ni is the number of vertices in Gi. Our main result is that there exist visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2n. An upper bound of 0(n2/log n) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of size O(n log3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).
39 citations
TL;DR: It is proved that it remains NP-complete when the input is restricted to the classes of split graphs or graphs with maximum degree 3 k +2, and it is shown that the SPANNING 2-TREE problem remains NP -complete when restricted to planar graphs withmaximum degree at most 6.
24 citations
TL;DR: In this article, the authors study matrices whose inverses exhibit a sparsity pattern that may be described with a chordal graph and derive an explicit entry-wise formula that is useful in certain matrix completion problems.
Abstract: We study matrices whose inverses exhibit a sparsity pattern that may be described with a chordal graph. Such matrices are characterized in terms of a special vanishing-minor structure, and an explicit entry-wise formula that is useful in certain matrix completion problems is derived.We also study an interesting graph-inheritance principle in the context of chordal graphs.
14 citations
TL;DR: It is able to show that (1 − c ) n 2 /4 cliques will suffice for some c > 0.
Abstract: To partition the edges of a chordal graph on n vertices into cliques may require as many as n 2 /6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c ) n 2 /4 cliques will suffice for some c > 0.
6 citations
TL;DR: In this paper, the structure of minimal vertex separators of 2-Chordal graphs is determined, and it is possible to choose a clique (complete subgraph) as a minimal node separator.
4 citations