Paired domination subdivision and multisubdivision numbers of graphs

Abstract

The paired domination subdivision number \( sd_p(G) \) of a graph \( G \) is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the paired domination number of \( G \). We prove that the decision problem of the paired domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the paired domination multisubdivision number of a nonempty graph \( G \), denoted by \( msd_p(G) \), as the minimum positive integer \( k \) such that there exists an edge which must be subdivided \( k \) times to increase the paired domination number of \( G \). We show that \( msd_p(G) \leq 4 \) for any graph \( G \) with at least one edge. We also determine paired domination multisubdivision numbers for some classes of graphs. Moreover, we give a constructive characterizations of all trees with paired domination multisubdivision number equal to 4.