On computing secure domination of trees

Abstract

Let [Formula: see text] be a graph. A subset [Formula: see text] is a dominating set of [Formula: see text] if for each [Formula: see text] there is a vertex [Formula: see text] adjacent to [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is a secure dominating set of [Formula: see text] if for each [Formula: see text] there is a vertex [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text] is also a dominating set of [Formula: see text]. The minimum cardinality of a secure dominating set of [Formula: see text] is called the secure domination number of [Formula: see text]. Burger et al. [A linear algorithm for secure domination in trees, Discrete Appl. Math. 171 (2014) 15–27] proposed a nontrivial algorithm for computing a minimum secure dominating set of a given tree in linear time and space. In this paper, we give a dynamic programming algorithm to compute the secure domination number of a given tree [Formula: see text] in [Formula: see text] time and space and then using a backtracking search algorithm we can find a minimum secure dominating set of [Formula: see text] in [Formula: see text] time and space that its implementation is much simpler than the implementation of the algorithm proposed by Burger et al.