Weighted domination of independent sets

Abstract

The {\em independent domination number} $\gamma^i(G)$ of a graph $G$ is the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$. It is known \cite{abz} that in chordal graphs $\gamma^i$ is equal to $\gamma$, the ordinary domination number. The weighted version of this result is not true, but we show that it does hold for interval graphs, and for the intersection (that is, line) graphs of subtrees of a given tree, where each subtree is a single edge.