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- W3028891342 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." @default.
- W3028891342 created "2020-06-05" @default.
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- W3028891342 date "2020-05-29" @default.
- W3028891342 modified "2023-09-24" @default.
- W3028891342 title "Which Graphs Occur as $$gamma $$-Graphs?" @default.
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- W3028891342 doi "https://doi.org/10.1007/s00373-020-02174-9" @default.
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