Paired-domination in inflated graphs

The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique Kd. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γp(G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G) ≥ 2, then n(G) ≤ γp(GI) ≤ 4m(G) δ(G)+1 , and the equality γp(GI) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.