Connected-domination triangle graphs, perfect connected-neighbourhood graphs and connected neighbourhood sets

Abstract

We introduce and characterize the class of graphs in which every connected dominating set is a (connected) neighbourhood set and the class of graphs whose all connected induced subgraphs have equal minimum neighbourhood set and minimum connected neighbourhood set cardinalities. Assuming P - NP, we also prove that the minimum connected neighbourhood set problem cannot be approximated within a logarithmic factor in polynomial time in their common subclass, the class of simplicial split graphs.