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K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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TL;DR: In this article, a particular partition of the vertex set of claw-free strongly chordal graphs in which each element is a clique was found, and it was shown that the adjacency graph of these cliques is a tree.
Abstract: In this paper we find a particular partition of the vertex set of claw-free strongly chordal graphs in which each element is a clique, and we show that the adjacency graph of these cliques is a tree. In particular, the presented results imply the existence of an ordering of the vertices, and a corresponding edge orientation, such that each directed path is contained in at most two maximal cliques. As shown by the authors in previous works, this allows to give performance guarantee approximation results on a wide class of optimization problems.
2 citations
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TL;DR: This work has succeeded in developing a fast algorithm for maximum clique problem by employing the method of verification and elimination and running in polynomial time when applied to random graphs and DIMACS benchmark graphs.
Abstract: A clique in an undirected graph G= (V, E) is a subset V' V of vertices, each pair of which is connected by an edge in E. The clique problem is an optimization problem of finding a clique of maximum size in graph. The clique problem is NP-Complete. We have succeeded in developing a fast algorithm for maximum clique problem by employing the method of verification and elimination. For a graph of size N there are 2N sub graphs, which may be cliques and hence verifying all of them, will take a long time. Idea is to eliminate a major number of sub graphs, which cannot be cliques and verifying only the remaining sub graphs. This heuristic algorithm runs in polynomial time and executes successfully for several examples when applied to random graphs and DIMACS benchmark graphs.
2 citations
TL;DR: In this paper, the authors compute the spectra of adjacency matrices of semi-regular polytopes and show that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.
Abstract: We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovasz and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.
2 citations
TL;DR: The relation between three parameters of a chordal graph G: the number of non-separating cliques nsc(G), the asteroidal number an(G) and the leafage l(G).
2 citations
TL;DR: A necessary and sufficient condition so that it is true, in terms of the minimal vertex separators of the graph, that the clique trees of G are exactly the compatible trees of K .
2 citations