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arXiv · 2303.06299

Reconfiguration of Minimum Independent Dominating Sets in Graphs

R. C. Brewster·C. M. Mynhardt·L. E. Teshima
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Abstract

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the $i(G)$-sets, and where two $i(G)$-sets are adjacent if and only if they differ by two adjacent vertices. We show that not all graphs are $i$-graph realizable, that is, given a target graph $H$, there does not necessarily exist a source graph $G$ such that $H$ is isomorphic to $\mathcal{I}(G)$. Examples of such graphs include $K_{4}-e$ and $K_{2,3}$. We build a series of tools to show that known $i$-graphs can be used to construct new $i$-graphs and apply these results to build other classes of $i$-graphs, such as block graphs, hypercubes, forests, cacti, and unicyclic graphs.

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R. C. Brewster, C. M. Mynhardt, L. E. Teshima. 2023-03-11. Reconfiguration of Minimum Independent Dominating Sets in Graphs. https://arxiv.org/abs/2303.06299

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