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

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 $H$.