The co-secure domination in proper interval graphs

A set S⊂V is a co-secure dominating set of a graph G=(V,E) if S is a dominating set, and for each u∈S there exists a vertex v∈V∖S such that uv∈E and (S∖{u})∪{v} is a dominating set. Note that |V|>1. The minimum cardinality of a co-secure dominating set in G is the co-secure domination number of G. In this paper, we propose a linear-time algorithm for finding the co-secure domination number of proper interval graphs.