Complexity of paired domination in AT-free and planar graphs

For a graph G=(V,E), a subset D of vertex set V, is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem is NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the other side, the problem is efficiently solvable for many graph classes including interval graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graphs. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NP-complete for planar graphs, which answers an open question asked by Lin et al. (2015) [19] and (2020) [20]. Alvarado et al. (2015) [1] conjectured that given a graph G=(V,E), it is NP-hard to decide whether γ(G)=γ_pr(G). In this paper we settle this conjecture affirmatively.