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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: It is proved that the recognition problem for coordinated graphs is NP-hard, and it isNP-complete even when restricted to the class of {gem, C4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.
Abstract: A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. In previous works, polynomial time algorithms were found for recognizing coordinated graphs within some classes of graphs. In this paper we prove that the recognition problem for coordinated graphs is NP-hard, and it is NP-complete even when restricted to the class of {gem, C4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.
8 citations
28 Jul 1996
TL;DR: Haynes and Schoenefeld as mentioned in this paper presented a genetic programming (GP) technique to find cliques in a graph, where a clique is defined as a complete subgraph of a graph.
Abstract: We present a genetic programming (GP) technique to find cliques in a graph [Haynes and Schoenefeld, 1996]. A collection of cliques in a graph can be represented as a list of a list of nodes which, in turn, can be represented by a tree structure. Given a graph G = (V, E) a clique of G is a complete subgraph of G. We denote a clique by the set of vertices in the complete subgraph. Our goal is to find all cliques of G. Since the subgraph of G induced by any subset of the vertices of a complete subgraph of G is also complete, it is sufficient to find all maximal complete subgraphs of G. A maximal complete subgraph of G is a maximal clique. Each S--expression in a GP pool will represent sets of candidate maximal cliques. The function and terminal sets are F = {ExtCon, IntCon} and T = {1, ..., #nodes}. ExtCon "separates" two candidate maximal cliques, while IntCon "joins" two candidate cliques to create a larger candidate.
8 citations
Posted Content•
TL;DR: It is shown that for any chordal graph the authors can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal under inclusion among all minimal separators.
Abstract: We present here some results on particular elimination schemes for chordal graphs, namely we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal (resp. minimal) under inclusion among all minimal separators.
8 citations
Proceedings Article•
06 Jan 2007
TL;DR: In this paper, a linear-time algorithm for recognizing and realizing degree sequences of 2-trees is presented, which yields a linear time algorithm for finding a 2-tree.
Abstract: A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbours are adjacent, and G \ v is a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees.
8 citations
TL;DR: A simple notion of strong clique trees is proposed in this paper for strongly chordal graphs, which can be extended to chordal bipartite graphs, clique graphs of chordal vertices, and incidence graphs of biacyclic hypergraphs.
Abstract: Maximal complete subgraphs and clique trees are basic to both the theory and applications of chordal graphs. A simple notion of strong clique tree extends this structure to strongly chordal graphs. Replacing maximal complete subgraphs with open or closed vertex neighborhoods discloses new relationships between chordal and strongly chordal graphs and the previously studied families of chordal bipartite graphs, clique graphs of chordal graphs (dually chordal graphs), and incidence graphs of biacyclic hypergraphs. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 151–160, 2000
8 citations