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 & SUBE; 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 & ISIN; V \ D there is a vertex u & ISIN; D adjacent to v such that (D \ {u}) & OR; {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 - epsilon)ln | V | for any epsilon > 0, unless NP & SUBE; DTIME(|V |(O (log log |V |))). (c) 2022 Elsevier B.V. All rights reserved.