Secure domination in proper interval graphs

A subset S of vertices in a graph G is a secure dominating set of G if S is a dominating set of G and, for each vertex u∉S, there is a vertex v∈S such that uv is an edge and (S∖{v})∪{u} is also a dominating set of G. The secure domination number γs(G) is the cardinality of a smallest secure dominating set of G. In this paper, we propose a linear-time algorithm for finding the secure domination number of proper interval graphs.