Partial domination and irredundance numbers in graphs

Abstract

A dominating set of a graph G=(V(G),E(G)) is a vertex set D such that every vertex in V(G)∖D is adjacent to a vertex in D. The cardinality of a smallest dominating set of G is called the domination number of G and is denoted by γ(G). A vertex set D is a k-isolating set of G if G−NG[D] contains no k-cliques. The minimum cardinality of a k-isolating set of G is called the k-isolation number of G and is denoted by ιk(G). Clearly, γ(G)=ι1(G). A vertex set I is irredundant if, for every non-isolated vertex v of G[I], there exists a vertex u in V(G)∖I such that NG(u)∩I={v}. An irredundant set I is maximal if the set I∪{u} is no longer irredundant for any u∈V(G)∖I. The minimum cardinality of a maximal irredundant set is called the irredundance number of G and is denoted by ir(G). Allan and Laskar [1] and Bollobás and Cockayne [2] independently proved that γ(G)<2ir(G), which can be written ι1(G)<2ir(G), for any graph G. In this paper, for a graph G with maximum degree Δ, we establish sharp upper bounds on ιk(G) in terms of ir(G) for Δ−2≤k≤Δ+1.