Unique irredundance, domination and independent domination in graphs

Abstract

A subset D of the vertex set of a graph G is irredundant if every vertex v in D has a private neighbor with respect to D, i.e. either v has a neighbor in V (G)\D that has no other neighbor in D besides v or v itself has no neighbor in D. An irredundant set D is maximal irredundant if D ∪ {v} is not irredundant for any vertex v ∈ V (G)\D. A set D of vertices in a graph G is a minimal dominating set of G if D is irredundant and every vertex in V (G)\D has at least one neighbor in D. A subset I of the vertex set of a graph G is independent if no two vertices in I are adjacent. Further, a maximal irredundant set, a minimal dominating set and an independent dominating set of minimum cardinality are called a minimum irredundant set, a minimum dominating set and a minimum independent dominating set, respectively, and the cardinalities of these sets are called the irredundance number, the domination number and the independent domination number, respectively.In this paper we prove that any graph with equal irredundance and domination numbers has a unique minimum irredundant set if and only if it has a unique minimum dominating set. Using a result by Zverovich and Zverovich [An induced subgraph characterization of domination perfect graphs, J. Graph Theory 20(3) (1995) 375-395], we characterize the hereditary class of graphs G such that for every induced subgraph H of G, H has a unique -set if and only if H has a unique -set. Furthermore, for trees with equal domination and independent domination numbers we present a characterization

Common domination perfect graphs

“…Generalizing the concept of domination perfect graphs, given any two graph parameters λ and µ for which λ(G) ≤ µ(G), a graph G is defined in [4,5,7,12] to be (λ, µ)-perfect if λ(H) = µ(H) for every induced subgraph H of G. In particular, a domination perfect graph is a (γ, i)-perfect graph. In this paper we study (γ, α c )-perfect graphs, that is, we study graphs G satisfying γ(H) = α c (H) for every induced subgraph H of G. We call such graphs common domination perfect graphs since here we study perfect graphs with respect to the domination number and the common independence number.…”

Trees with unique minimum global offensive alliance

“…Several works on uniqueness related to other graph parameters have been widely studied, such as locating-domination number [1], paired-domination number [3], double domination number [4], roman domination number [5] and total domination number [17]. Further work on this topic can be found in [8,10,11,12,16,21,22,23] The aim of this paper is to characterize all trees having unique minimum global offensive alliance set. We denote such trees as UGOA-trees.…”