Which graphs occur as $\gamma$-graphs?
| Autor: | DeVos, Matt, Dyck, Adam, Jedwab, Jonathan, Simon, Samuel |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 28 pages, 5 figures, 2 appendices. Simplified proof of Theorem 1.5 and some minor revisions |
| Databáze: | arXiv |
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