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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: The effect of operations like edge contraction, edge removal and others on the dynamical behavior of a graph under the iteration of the clique operator K is explored and it is proved that every clique Divergent graph is a spanning subgraph of a clique divergent graph with diameter two.
4 citations
TL;DR: A defining linear system of STAB (G) and CLIQUE(G) is given when G is a graph containing no P"5 and no gem.
4 citations
Dissertation•
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31 Dec 2014
TL;DR: In this article, it was shown that the complete 3-partite graph K1,n,m is the graph of a polytope if and only if Kn,n is the vertex set of K 1,m.
Abstract: The graph of a polytope is the graph whose vertex set is the set of vertices of the polytope, and whose edge set is the set of edges of the polytope. Several problems concerning graphs of polytopes are discussed. The primary result is a set of bounds (Theorem 39) on the maximal size of an anticlique (sometimes called a coclique, stable set, or independent set) of the graph of a polytope based on its dimension and number of vertices. Two results concerning properties preserved by certain operations on polytopes are presented. The first is that the Gale diagram of a join of polytopes is the direct sum of the Gale diagrams of the polytopes and dually, that the Gale diagram of a direct sum of polytopes is the join of their Gale diagrams (Theorem 23). The second is that if two polytopes satisfy a weakened form of Gale’s evenness condition, then so does their product (Theorem 32). It is shown, by other means, that, with only two exceptions, the complete bipartite graphs are never graphs of polytopes (Theorem 47). The techniques developed throughout are then used to show that the complete 3-partite graph K1,n,m is the graph of a polytope if and only if Kn,m is the graph of a polytope (Theorem 49). It is then shown that K2,2,3 and K2,2,4 are never graphs of polytopes. A conjecture is then stated as to precisely when a complete multipartite graph is the graph of a polytope. Finally, a section is devoted to results concerning the dimensions for which the graph of a crosspolytope is the graph of a polytope.
4 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
28 May 2014
TL;DR: It is proved that the number of maximal cliques does not exceed n k, and it is shown that this bound is tight for any fixed k.
Abstract: Since an image can easily be modeled by its adjacency graph, graph theory and algorithms on graphs are widely used in image processing. Of particular interest are the problems of estimating the number of the maximal cliques in a graph and designing algorithms for their computation, since these are found relevant to various applications in image processing and computer graphics. In the present paper we study the maximal clique problem on intersection graphs of convex polygons, which are also applicable to imaging sciences. We present results which refine or improve some of the results recently proposed in [18]. Thus, it was shown therein that an intersection graph of n convex polygons whose sides are parallel to k different directions has no more than n 2k maximal cliques. Here we prove that the number of maximal cliques does not exceed n k . Moreover, we show that this bound is tight for any fixed k. Algorithmic aspects are discussed as well.
4 citations