K-tree

About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.

Clique-transversal sets and clique-coloring in planar graphs

Abstract: Let G=(V,E) be a graph. A clique-transversal setD is a subset of vertices of G such that D meets all cliques of G, where a clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. The clique-transversal number, denoted by @t"C(G), of G is the cardinality of a minimum clique-transversal set in G. A k-clique-coloring of G is a k-coloring of its vertices such that no clique is monochromatic. All planar graphs have been proved to be 3-clique-colorable by Mohar and Skrekovski [B. Mohar, R. Skrekovski, The Grotzsch theorem for the hypergraph of maximal cliques, Electron. J. Combin. 6 (1999) #R26]. Erdos et al. [P. Erdos, T. Gallai, Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math. 108 (1992) 279-289] proposed to find sharp estimates on @t"C(G) for planar graphs. In this paper we first show that every outerplanar graph G of order n(>=2) has @t"C(G)@?3n/5 and the bound is tight. Secondly, we prove that every claw-free planar graph different from an odd cycle is 2-clique-colorable and we present a polynomial-time algorithm to find the 2-clique-coloring. As a by-product of the result, we obtain a tight upper bound on the clique-transversal number for claw-free planar graphs.

The Validity of the Perfect Graph Conjecture for K4-Free Graphs

Abstract: Publisher Summary This chapter builds on results based on D.R. Fulkerson's anti-blocking polyhedral approach to perfect graphs to obtain information about critical perfect graphs and related clique-generated graphs. Fulkerson felt that a proof of the perfect graph theorem would involve exactly the kind of duality that existed in his theory of blocking and anti-blocking polyhedral. A critical perfect graph—p-critical for short—is an imperfect graph all of whose proper induced subgraphs are perfect. A p-critical graph with n vertices has exactly n cliques of size ω (G) with each vertex in ω (G) maximal cliques and has exactly n stable sets of size α (G) with each vertex in α (G) maximal stable sets. Each maximal clique intersects all but one maximal stable sets, and vice versa. If G is a pseudo-p-critical graph, each maximal clique in M (G) corresponds to a vertex of G.

On clique convergent graphs

Abstract: A graphG isconvergent when there is some finite integern ? 0, such that then-th iterated clique graphK n(G) has only one vertex. The smallest suchn is theindex ofG. TheHelly defect of a convergent graph is the smallestn such thatK n(G) is clique Helly, that is, its maximal cliques satisfy the Helly property. Bandelt and Prisner proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the difference of its index and diameter by more than one. In the present paper an affirmative answer to the question is given. For any arbitrary finite integern, a graph is exhibited, the Helly defect of which exceeds byn the difference of its index and diameter.