Pairwise and variance balanced n-ary block designs

Nutan Mishra1, Dinesh G. Sarvate1
1Faculty of Applied Physics and Mathematics Gdańsk University of Technology, ul. Narutowicza 11/12, 80-233 Gdańsk, Poland

Abstract

The paired domination subdivision number \( \text{sd}_{pr}(G) \) 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 \textit{paired domination multisubdivision number} of a nonempty graph \( G \), denoted by \( \text{msd}_{pr}(G) \), 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 \( \text{msd}_{pr}(G) \leq 4 \) 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 characterization of all trees with paired domination multisubdivision number equal to 4.

Keywords: Paired domination; domination subdivision number; domination multisubdivision number; computational complexity; trees.