A note on some variations of the $\gamma$-graph

TL;DR

This paper explores various modifications of the $\gamma$-graph, demonstrating that for any graph $H$, infinitely many graphs can be constructed whose $\gamma$-graph variants are isomorphic to $H$, thus revealing the versatility of these graph constructs.

Contribution

The paper introduces several variations of the $\gamma$-graph, including those based on identifying codes, locating-domination, and other domination parameters, and proves their universality.

Findings

For any graph $H$, infinitely many graphs have a $\gamma$-graph variant isomorphic to $H$.

Multiple $\gamma$-graph variations are introduced and analyzed.

The universality of these variants is established across different domination concepts.

Abstract

For a graph $G$, the $\gamma$-graph of $G$, $G(\gamma)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(\gamma)$ are adjacent if and only if their corresponding dominating sets in $G$ differ by exactly two adjacent vertices. In this paper, we present several variations of the $\gamma$-graph including those using identifying codes, locating-domination, total-domination, paired-domination, and the upper-domination number. For each, we show that for any graph $H$, there exist infinitely many graphs whose $\gamma$-graph variant is isomorphic to $H$.