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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: A sufficient condition for a graph to have a k-tree containing specified vertices as following: let G be a connected graph and S be a subset of V(G).
Abstract: A tree T is called a k-tree if the maximum degree of T is at most k. In this paper, we give a sufficient condition for a graph to have a k-tree containing specified vertices as following: let G be a connected graph and let S be a subset of V(G). If \(\alpha _G(S)\le (k-1)\kappa _G(S)+1\), then G has a k-tree containing S. Moreover, this condition is sharp.
2 citations
26 May 2008
TL;DR: Along the way, an open question in the literature on the maxcut problem is answered, by showing that the so-called k-gonal inequalities define a polytope.
Abstract: We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and well-known integer polytopes -- the cut, boolean quadric, multicut and clique partitioning polytopes -- are shown to arise as projections of binary psd polytopes. Finally, we present various valid inequalities for binary psd polytopes, and show how they relate to inequalities known for the simpler polytopes mentioned. Along the way, we answer an open question in the literature on the maxcut problem, by showing that the so-called k-gonal inequalities define a polytope.
2 citations
TL;DR: In this article, all minimal clique separators of all four standard products: Cartesian, the strong, the direct, and the lexicographic, as well as all maximal atoms of the Cartesian and the strong products were described.
Abstract: We describe in present work all minimal clique separators of all four standard products: the Cartesian, the strong, the direct, and the lexicographic, as well as all maximal atoms of the Cartesian, the strong and the lexicographic product, while we only partially describe maximal atoms of the direct product. In most cases a product have no clique separator and is thus the hole product the maximal atom.
2 citations
TL;DR: It is proved that a self-complementary graph with p vertices is chordal if and only if its clique number is integral part of ( p + 1)/2.
2 citations
TL;DR: A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes and a generalization of Haberdasher’s Puzzle are presented.
Abstract: ・ Reversibility and Foldability of Conway Tiles (with K. Matsunaga), Computational Geometry, Vol.64 (2017), 30-45 ・ A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes (with Sin Hitotumatu, Motonaga Ishii, Akihiro Matuura, Ikuro Sato and Shun Toyoshima), Journal of Information Processing Vol.25, 528-536 (2017) ・ Generalization of Haberdasher’s Puzzle (with Kiyoko Matsunaga), Discrete and Computational Geometry Vol.58 Issue 1, 30-50 ・ Reversible Nets of Polyhedra (with S. Langerman and K. Matsunaga), Discrete and Computational Geometry and Graphs 2015, LNCS9943 (2016),13-23
2 citations