On independent domination in direct products

TL;DR

This paper disproves a conjecture about the independent domination number in direct graph products and provides exact values for specific cases, advancing understanding of graph product properties.

Contribution

It shows the conjecture that $i(G imes H) \\ge i(G)i(H)$ is false and constructs counterexamples with arbitrarily large differences, also determining $i(G imes K_n)$ for paths and cycles.

Findings

Counterexamples with large differences in independent domination numbers

Disproof of the conjecture $i(G imes H) \\ge i(G)i(H)$

Exact values of $i(G imes K_n)$ for paths and cycles

Abstract

In \cite{nr-1996} Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that $i(G\times H) \ge i(G)i(H)$ where $i(G)$ is the independent domination number of $G$ and $G\times H$ is the direct product of graphs $G$ and $H$. We show this conjecture is false, and, in fact, construct pairs of graphs for which $\min\{i(G), i(H)\} - i(G\times H)$ is arbitrarily large. We also give the exact value of $i(G\times K_n)$ when $G$ is either a path or a cycle.