Irregular Domination Trees and Forests

A set S of vertices in a connected graph G is an irregular dominating set if the vertices of S can be labeled with distinct positive integers in such a way that for every vertex v of G , there is a vertex u ∈ S such that the distance from u to v is the label assigned to u . If for every vertex u ∈ S , there is a vertex v of G such that u is the only vertex of S whose distance to v is the label of u , then S is a minimal irregular dominating set. A graph H is an irregular domination graph if there exists a graph G with a minimal irregular dominating set S such that H is isomorphic to the subgraph G [ S ] of G induced by S . In this paper, all irregular domination trees and forests are characterized. All disconnected irregular domination graphs are determined as well.