Adjoining edges to $G\mathbin{\square}H$ to construct a minimal dominating set of size $\gamma(G)\gamma(H)$

TL;DR

The paper explores how adding edges to the Cartesian product of two graphs can produce a minimal dominating set of a specific size, potentially implying Vizing's conjecture.

Contribution

It introduces a method to add edges to $G imes H$ to create a minimal dominating set of size $eta(G)eta(H)$ and links this to Vizing's conjecture.

Findings

Adding edges can produce a minimal dominating set of size $eta(G)eta(H)$

The approach implies Vizing's conjecture if the set is also minimum

Provides a new perspective on dominating sets in graph products

Abstract

For graphs $G,H$ it is possible to add $(|V(G)|-\gamma(G))(|V(H)|-\gamma(H))$ edges to the Cartesian product $G\mathbin{\square}H$ such that a minimal dominating set $D$ of size $\gamma(G)\gamma(H)$ emerges. We hypothesize that $D$ is also a minimum dominating set for the resulting graph and show that this implies Vizing's conjecture.