Upper Bounds on the Paired Domination Subdivision Number of a Graph

A paired dominating set of a graph G with no isolated vertex is a dominating set S of vertices such that the subgraph induced by S in G has a perfect matching. The paired domination number of G, denoted by γpr(G), is the minimum cardinality of a paired dominating set of G. The paired domination subdivision number $${{\rm sd}_{\gamma _{\rm pr}}(G)}$$ is the minimum number of edges to be subdivided (each edge of G can be subdivided at most once) in order to increase the paired domination number. In this paper, we show that if G is a connected graph of order at least 4, then $${{\rm sd}_{\gamma _{\rm pr}}(G)\leq 2|V(G)|-5}$$. We also characterize trees T such that $${{\rm sd}_{\gamma _{\rm pr}}(T) \geq |V(T)| /2}$$.