Complexity of paired domination in AT-free and planar graphs

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