Weighted domination of independent sets

TL;DR

This paper investigates the weighted independent domination number in specific graph classes, showing it equals the standard domination number in interval graphs and certain subtree intersection graphs, extending known results from chordal graphs.

Contribution

It proves that the weighted independent domination number equals the domination number for interval graphs and certain subtree intersection graphs, generalizing previous chordal graph results.

Findings

Weighted independent domination number equals domination number in interval graphs.

This equality also holds for intersection graphs of subtrees of a tree with single-edge subtrees.

Extends known results from chordal graphs to broader classes.

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.