A Note on the Paired-Domination Subdivision Number of Trees

For a graph G with no isolated vertex, let gamma(pr)(G) and sd(gamma pr)(G) denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order n >= 4 different from a healthy spider (subdivided star), then sd(gamma pr)(T) <= min{gamma(pr)(T)/2 + 1, n/2}, improving the (n-1)-upper bound that was recently proven.

Automated summary

In this note, it is shown that for a tree different from a healthy spider, the paired-domination subdivision number satisfies a certain condition, improving the existing upper bound.