Complexity Issues of Variants of Secure Domination in Graphs

An improved binary programming formulation for the secure domination problem

“…It is clear that equations (7) and ( 12) satisfy requirement 1. It can also been seen that equations ( 8), (10) and ( 11) satisfy requirement 2, by the following argument. First, constraints (8) and (11) ensure that if x i = 0, then y ij = 0.…”

“…Hence, there must be a guard at vertex i in order to choose it as a designated guard for another vertex. Then, for each vertex j, constraint (10) ensures that either x j = 1 or i∈N (i) y ij = 1; this corresponds precisely to requirement 2.…”

“…Since requirement 2 is satisfied, there are two possibilities; either there is a guard present at vertex j, or vertex j has at least one designated guard at some other vertex k. Consider first the case where there is a guard present at vertex j; clearly, no designated guard needs to move in this case. Then, by constraint (10) it is clear that y kj = 0 for all k, and hence constraint (9) reduces to constraint (7).…”

“…In the short time since the concept of secure connected domination was introduced, a small number results have been obtained. In particular, some bounds and exact values of the secure connected domination numbers have been found for graphs resulting from joins and compositions of graphs were given by Cabaro and Canoy Jr. [5], while Lad, Reddy and Kumar [10] proved that the secure connected domination problem is NP-complete in the case of split graphs. Here, we add to the literature on this problem by providing a binary programming formulation for the secure connected domination problem.…”