Parameterized Complexity of Secure Sets

Secure sets were introduced by Brigham, Dutton, and Hedetniemi [1] as an extension of defensive alliances in a graph. Members of a defensive alliance have agreed to come to the aid of their neighbors in the alliance for mutual protection from attack by nonmembers. Alliances can occur, for example, between nations, biological sequences, and business cartels, and also have been used to model a variety of other applications such as classification problems, communities in the World Wide Web, etc [7, 9]. Let G = (V,E), be a graph with |V | = n vertices. For a vertex x ∈ V , the open neighborhood of x is N(x) = {y ∈ V : xy ∈ E}, the degree of x is dx = |N(x)|, and the closed neighborhood of x is N [x] = N(x) ∪ {x}. Similarly, for any X ⊆ V , the closed neighborhood of X is N [X] = ⋃ x∈X N [x], and the boundary of S ⊆ V , ∂S, is N [S]− S. Other graph related terms and definitions can be found in [2]. A defensive alliance in a graph G = (V,E) is a non empty set S ⊆ V , where for all x ∈ S, |N [x] ∩ S| ≥ |N [x] − S|, that is, for every x ∈ S, at least bdx/2c neighbors of x are also members of S. If S is a defensive alliance and x ∈ S, the vertices in N [x] ∩ S can defend an attack from vertices in N [x] − S. Consequently, every vertex x that is a member of a defensive alliance has at least as many vertices defending it (including itself) as there are vertices attacking it. Therefore, an attack on a vertex in a defensive alliance can be neutralized by its defenders. While a single vertex in a defensive alliance can be defended, a coordinated attack on multiple members may not be defendable. Secure sets address this weakness. A set S is said to be secure if |N [X] ∩ S| ≥ |N [X]− S|, for every X ⊆ S. This condition was shown in [1] to be necessary and sufficient for defending a simultaneous attack on any set of vertices in a secure set. To date, there is no known polynomial time algorithm for identifying a secure set in a graph. Indeed, there is not even a known polynomial time algorithm for determining whether a given set S is secure. In this paper we present a fixed-parameter tractable algorithm (FPT-algorithm) for the secure set problem. In Section 2, an introduction to the theory of parameterized complexity and known results on the complexity of the decision problems on secure sets and defensive alliances are given. Section 3 presents lemmas to show the correctness of the FPT-algorithm given in Section 4, followed by an analysis of the complexity of the algorithm in Section 5, and conclusions in Section 6.