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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: Characterizations of graphs whose total-block graphs are maximal outerplanar or maximal minimally nonouterplanar are presented and it is proved that for any nontrivial graph G,TnB(G) is not maximal outer Planar for all n ≥ 2.
31 Mar 2014
TL;DR: This work solves an open problem posed by Kratochvil and Tuza to determine the complexity of 2-clique-colouring of perfect graphs with all cliques having size at least 3, and determines a hierarchy of nested subclasses of weakly chordal graphs whereby each graph class is in a distinct complexity class.
Abstract: A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A k-clique-colouring of a graph is a colouring of the vertices with at most k colours such that no clique is monochromatic. Defossez proved that the 2-clique-colouring of perfect graphs is a \(\Sigma_2^P\)-complete problem [J. Graph Theory 62 (2009) 139–156]. We strengthen this result by showing that it is still \(\Sigma_2^P\)-complete for weakly chordal graphs. We then determine a hierarchy of nested subclasses of weakly chordal graphs whereby each graph class is in a distinct complexity class, namely \(\Sigma_2^P\)-complete, \(\mathcal{NP}\)-complete, and \(\mathcal{P}\). We solve an open problem posed by Kratochvil and Tuza to determine the complexity of 2-clique-colouring of perfect graphs with all cliques having size at least 3 [J. Algorithms 45 (2002), 40–54], proving that it is a \(\Sigma_2^P\)-complete problem. We then determine a hierarchy of nested subclasses of perfect graphs with all cliques having size at least 3 whereby each graph class is in a distinct complexity class, namely \(\Sigma_2^P\)-complete, \(\mathcal{NP}\)-complete, and \(\mathcal{P}\).
TL;DR: It is shown that every connected K1,k+1-free graph G has a spanning k-tree if the degree sum of any 3k − 3 independent vertices in G is at least |G | − 2, where |G| is the order of G.
Abstract: Abstract For an integer k ≥ 2, a k-tree T is defined as a tree with maximum degree at most k. If a k-tree T spans a graph G, then T is called a spanning k-tree of G. Since a spanning 2-tree is a Hamiltonian path, a spanning k-tree is an extended concept of a Hamiltonian path. The first result, implying the existence of k-trees in star-free graphs, was by Caro, Krasikov, and Roditty in 1985, and independently, Jackson and Wormald in 1990, who proved that for any integer k with k ≥ 3, every connected K1,k-free graph contains a spanning k-tree. In this paper, we focus on a sharp condition that guarantees the existence of a spanning k-tree in K1,k+1-free graphs. In particular, we show that every connected K1,k+1-free graph G has a spanning k-tree if the degree sum of any 3k − 3 independent vertices in G is at least |G| − 2, where |G| is the order of G.
15 Mar 2012
TL;DR: In this article, a new clustering approach was presented, using gene interaction graphs to model this data, and decomposing the graphs by means of clique minimal separators, where clique separators are a clique whose removal increases the number of connected components of the graph.
Abstract: The study of gene interactions is an important research area in biology and grouping genes with similar expression profiles to clusters is a first step towards a better understanding of their functional relationships. In Kaba et al. 2007, a new clustering approach was presented, using gene interaction graphs to model this data, and decomposing the graphs by means of clique minimal separators. A clique separator is a clique whose removal increases the number of connected components of the graph; the decomposition is obtained by repeatedly copying a clique separator into the components it defines, until only subgraphs with no clique separators are left: these subgraphs will be our clusters. The advantage of our approach is that this decomposition can be computed efficiently, is unique, and yields overlapping clusters. For that, the similarity between each pair of genes is estimated by a distance function, then a family of gene interaction graphs is constructed by choosing several thresholds, where an edge is added between two genes if their distance is below the threshold. Hereby, both the choice of the distance function and of the threshold influences the construction of the gene interaction graphs. In Kaba et al. 2007, several criteria are developed to select thresholds in an appropriate way. Here we discuss the impact of the choice of the distance function; our results suggest that this choice does not effect the final decomposition of the gene interaction graphs into clusters.
16 Feb 2017
TL;DR: Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention and are extensions of the classical Symmetric Traveling Salesman Problem poly topes whose graphs contain the TSP polytope graphs as spanning subgraphs.
Abstract: Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex transitive, not even regular.