Pairing the vertices of dominating sets in graphs: algorithmic and hardness results

Abstract

Domination and its variations are widely studied graph optimization problems. In a graph G = (V;E), a set D V is called a dominating set if every vertex not in D has a neighbour in D. The MI N IMUM D OMI NAT I O N problem requires to compute a minimum cardinality dominating set of G. In this thesis, we studied the algorithmic results for five variants of domination. The variants of domination studied in this thesis are based on pairing each vertex of a dominating set with some vertex at distance at most 2. The variants of domination studied in the thesis are (i) paired domination, (ii) semipaired domination, (iii) disjunctive paired domination, (iv) semitotal domination, and (v) defensive domination. Let G = (V;E) be a graph and D be a subset of V . Then D is called (i) a paired dominating set if D is a dominating set and G[D] has a perfect matching; (ii) a semipaired dominating set if D is a dominating set and D can be partitioned into 2 elements subsets such that distance between vertices in each two element set is at most 2; (iii) a disjunctive paired dominating set if every vertex v =2 D is either adjacent to a vertex of D or there are two vertices in D at distance two from v in G and G[D] has a perfect matching; (iv) a semitotal dominating set if D is a dominating set and for each vertex x 2 D, there exists another vertex y 2 D such that the distance between x and y is at most 2 in G; (v) a k-defensive dominating set if for a positive integer k, any k distinct vertices of V can be mapped to k distinct vertices of D, so that each vertex v is mapped to a vertex in the closed neighbourhood of v. Unfortunately, all these five variants of domination are NP-hard for general graphs and some restricted graph classes. Some of these variants are well studied in the literature, but there are many gaps which need filling. In this thesis, we obtain various algorithmic results for these variants of domination. For each of these variants, we identify graph classes where these problems remain NP-hard and designed efficient algorithms for these problems in some graph classes. We also study the approximation algorithm and approximation hardness of these problems and obtain various results.