Connection between conjunctive capacity and structural properties of graphs

Abstract

Please cite this article in press as: M. Chlebík, J. Chlebí ková, Connection between conjunctive capacity and structural properties of graphs, Theor. Comput. Sci. (2014), http://dx.doi.org/10.1016/j.tcs. 2014.04.035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThe investigation of the asymptotic behaviour of various graph parameters in powers of a graph is motivated by problems in information theory and extremal combinatorics. Considering various parameters and/or various notions of graph powers we can arrive at different notions of graph capacities, of which the Shannon capacity is best known. Here we study a related notion of the so-called conjunctive capacity of a graph G, C AND (G), introduced and studied by Gargano, Körner and Vaccaro. To determine C AND (G) is a convex programming problem. We show that the optimal solution to this problem is unique and describe the structure of the solution in any (simple) graph. We prove that its reciprocal value vc C (G):is an optimal solution of the newly introduced problem of Minimum Capacitary Vertex Cover that is closely related to the LPrelaxation of the Minimum Vertex Cover Problem. We also describe its close connection with the binding number/binding set of a graph, and with the strong crown decomposition of graphs.