Contributions to the theory of domination, independence and irredundance in graphs

Abstract

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X−{ x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X−{ x}. The lower and upper irredundance numbers ir( G) and IR( G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ( G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i( G) and the independence number β( G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G. A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely i+β⩽2p + 2δ - 2 2pδ , is proved.