Vizing's conjecture for cographs

TL;DR

This paper proves Vizing's conjecture for the class of cographs, showing that the domination number of their Cartesian product with any graph is at least the product of their individual domination numbers, using novel techniques.

Contribution

The paper introduces new methods to prove Vizing's conjecture specifically for cographs, expanding the class of graphs for which the conjecture holds.

Findings

Vizing's conjecture holds for cographs.

New proof techniques introduced for domination problems.

Results may extend to other graph classes.

Abstract

We show that if $G$ is a cograph, that is $P_4$-free, then for any graph $H$, $\gamma(G\square H)\geq \gamma(G)\gamma(H)$. By the characterization of cographs as a finite sequence of unions and joins of $K_1$, this result easily follows from that of Bartsalkin and German. However, the techniques used are new and may be useful to prove other results.