K-tree

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

Clique Complex Homology: A Combinatorial Invariant for Chordal Graphs

Abstract: It is shown that a geometric realization of the clique complex of a connected chordal graph is homologically trivial and as a consequence of this it is always the case for any connected chordal graph G that ∑_(k=1)^I‰(G)â–’(-1)^(k-1) I·_k (G)=1, where I·_k (G) is the number of cliques of order k in G and I‰(G) is the clique number of G.

Compressed Cliques Graphs, Clique Coverings and Positive Zero Forcing

Abstract: Zero forcing parameters, associated with graphs, have been studied for over a decade, and have gained popularity as the number of related applications grows. In particular, it is well-known that such parameters are related to certain vertex coverings. Continuing along these lines, we investigate positive zero forcing within the context of certain clique coverings. A key object considered here is the compressed cliques graph. We study a number of properties associated with the compressed cliques graph, including: uniqueness, forbidden subgraphs, connections to Johnson graphs, and positive zero forcing.

On the Complexity of fkg-domination for Chordal Graphs

Abstract: Due to its large range of applications, many variations and extensions of the classical domination problem in graphs have been defined and studied during the past fourty years. Given a graph G = (V,E), A ⊆ R and B = {b1, . . . , b|V |}, an A,B-dominating function of G is a function f : V 7→ A such that f(N [vi]) ≥ bi for all v ∈ V , where f(U) = ∑ u∈U f(u), for U ⊆ V and N [v] is the closed neighborhood of v. The weigth of f is given by w(f) = f(V ). This work is focused in two variations of the problem. Let k ∈ Z+ and bi = k for all i ∈ {1, . . . , |V |}. When A = {0, 1}, f is a k-tuple dominating function and γ×k(G) is the k-tuple domination number of G [3]. When A = {0, 1, . . . , k}, f is a {k}-dominating function and γ{k}(G) is the {k}-domination number of G [1]. As usual, these definitions induce the study of the following decision problems, for fixed k ∈ Z+: k-TUPLE DOMINATING FUNCTION (k-DOM) Instance: G = (V,E), j ∈ N Question: Does G have a k-tuple dominating function of weight at most j? {k}-DOMINATING FUNCTION ({k}-DOM) Instance: G = (V,E), j ∈ N Question: Does G have a {k}-dominating function of weight at most j? In this work we obtain a new graph class where {k}-DOM is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. By relating this problem with k-DOM, we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Secondorder Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. In this way we enlarge the family of graphs for which k-DOM is polynomial time solvable (see [2]).

On the arrangement of cliques in chordal graphs with respect to the cuts

Abstract: A cut (A, B) (where B = V - A) in a graph G = (V, E) is called internal if and only if there exists a vertex x in A that is not adjacent to any vertex in B and there exists a vertex y is an element of B such that it is not adjacent to any vertex in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A, B) in a chordal graph G, there exists a clique with kappa(G) + vertices (where kappa(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A, B). In fact we give a stronger result: For any internal cut (A, B) of a chordal graph, and for each i, 0 <= i <= kappa(G) + 1 such that vertical bar K-i vertical bar = kappa(G) + 1, vertical bar A boolean AND K-i vertical bar = i and vertical bar B boolean AND K-i vertical bar = kappa(G) + 1 - i. An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be Omega(k(2)), where kappa(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least kappa(G)(kappa(G)+1)/2 where kappa(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to kappa(G). This result is tight.

On Topological Indices And Domination Numbers Of Graphs

Abstract: Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distancebased topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index(PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well.