Algorithmic results in secure total dominating sets on graphs

Given a graph G=(V,E), a (total) dominating set of G is a subset D⊆V such that each vertex in V∖D (respectively, V) is adjacent to at least one vertex in D. A (total) dominating set D of G is a secure (total) dominating set if for each v∈V∖D there is a vertex u∈D adjacent to v such that (D∖{u})∪{v} is also a (total) dominating set of G. The minimum cardinality of a secure (total) dominating set of G is called the secure (total) domination number of G. The secure (total) domination problem is to find a minimum secure (total) dominating set of any graph. In this paper, we study the secure total domination problem. We show that the problem restricted to trees is solvable in linear-time, the decision version of the problem is NP-complete for circle graphs, the complexity of the problem differs from that of the secure domination problem, and the problem is APX-complete for graphs of degree at most 4. Furthermore, we show that the optimization version of the problem on bipartite graphs cannot be approximated in polynomial time within (1−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME(|V|O(log⁡log⁡|V|)).