Complexity and Algorithms for Semipaired Domination in Graphs

Abstract

For a graph G = (V, E) with no isolated vertices, a set D ⊆ V is called a semipaired dominating set of G if (i) D is a dominating set of G, and (ii) D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted by γ pr2 (G). The MINIMUM SEMIPAIRED DOMINATION problem is to find a semipaired dominating set of G of cardinality γ pr2 (G). In this paper, we initiate the algorithmic study of the MINIMUM SEMIPAIRED DOMINATION problem. We show that the decision version of the MINIMUM SEMI-PAIRED DOMINATION problem is NP-complete for bipartite graphs and split graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs and trees. We also propose a 1 + ln(2∆ + 2)-approximation algorithm for the MINIMUM SEMIPAIRED DOMINATION problem, where ∆ denote the maximum degree of the graph and show that the MINIMUM SEMIPAIRED DOMINATION problem cannot be approximated within (1 − ) ln |V | for any > 0 unless NP ⊆ DTIME(|V | O(log log |V |) ).

Unique minimum semipaired dominating sets in trees

“…In other words, the vertices in the dominating set S can be partitioned into 2-sets such that if {u, v} is a 2-set, then uv ∈ E(G) or the distance between u and v is 2. We say that u and v are paired, and that u and v are partners with respect to the resulting semi-matching consisting of the pairings of vertices of S. The semipaired domination number, denoted by γ pr2 (G), is the minimum cardinality of a semi-PD-set of G. We call a semi-PD-set of G of cardinality γ pr2 (G) a γ pr2 -set of G. Semipaired domination was introduced in [11] and studied, for example, in [12,13,[19][20][21]. From the definitions, we observe the following.…”