Irregular domination graphs

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 u of G , there is a vertex v ∈ S such that the distance from u to v is the label assigned to v . If for every vertex v ∈ S , there is a vertex u of G such that v is the only vertex of S whose distance to u is the label of v , 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 . We determine all paths and cycles that are irregular domination graphs as well as the familiar graphs of ladders and prisms, which are Cartesian products of K 2 with paths and cycles, respectively. Other results and problems are also presented on this topic.