Domination in Sparse Graphs

Abstract

Given a partition pi of V (G) with parts ( V1, V2, ... , V t). A pi-dominating set B is a dominating set that is the union of parts of pi. The pi-domination number gamma(G,pi) of G is the size of a smallest pi-dominating set. If each Vi in pi has size at most 2, we call pi a coupling of G and say that the vertices in Vi are coupled together. The coupled domination number, gamma cpl(G), is the maximum of gamma(G,pi) over all couplings pi of G. We also study coupled domination in trees. This topic was introduced by Slater and Seo, who established the coupled domination number of several different kinds of trees. They obtained a tree T' with gammacpl( T') = 8 and gamma(T' ) = 5. We show that gcplT gT ≥8/5 for every tree T with at least 2 vertices.