Which graphs occur as $\gamma$-graphs?

TL;DR

This paper investigates which graphs can be represented as $ ext{gamma}$-graphs, extending the concept to distance-$d$-domination, and characterizes such graphs among specific families like wheels, fans, and small graphs.

Contribution

It generalizes the notion of $ ext{gamma}$-graphs to distance-$d$-domination and provides a complete characterization for certain graph families.

Findings

The answer depends on whether the vertices admit a compatible labelling.

An explicit construction for graphs with prescribed minimum distance-$d$-dominating sets.

Complete characterization among wheel, fan, and small graphs.

Abstract

The $\gamma$-graph of a graph $G$ is the graph whose vertices are labelled by the minimum dominating sets of $G$, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size $\gamma(G)$) intersect in a set of size $\gamma(G)-1$. We extend the notion of a $\gamma$-graph from distance-1-domination to distance-$d$-domination, and ask which graphs $H$ occur as $\gamma$-graphs for a given value of~$d \ge 1$. We show that, for all $d$, the answer depends only on whether the vertices of $H$ admit a labelling consistent with the adjacency condition for a conventional $\gamma$-graph. This result relies on an explicit construction for a graph having an arbitrary prescribed set of minimum distance-$d$-dominating sets. We then completely determine the graphs that admit such a labelling among the wheel graphs, the fan graphs, and the graphs on at most six vertices. We connect the question of whether a graph admits such a labelling with previous work on induced subgraphs of Johnson graphs.