Unique irredundance, domination and independent domination in graphs

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 [email protected]?{v} is not irredundant for any vertex [email protected]?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 @i-set if and only if H has a unique @c-set. Furthermore, for trees with equal domination and independent domination numbers we present a characterization of unique minimum independent dominating sets, which leads to a linear time algorithm to decide whether such trees have unique minimum independent dominating sets.