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Independent domination versus minimum maximal matching in regular graphs

Prove that for every r-regular finite simple undirected graph G with r ≥ 3, the independent domination number i(G) satisfies i(G) ≤ μ*(G), where μ*(G) denotes the minimum cardinality of a maximal matching; moreover, demonstrate that this bound is sharp.

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Background

This problem mirrors a previously proven TxGraffiti-originated inequality for regular graphs, namely α(G) ≤ μ(G), by replacing maximum-size packings with their saturation analogues: the independent domination number i(G) and the minimum size of a maximal matching μ*(G).

The authors note the statement holds trivially for 2-regular graphs, concentrating the open case on regular graphs with degree at least 3. The conjecture reflects a symmetry between vertex- and edge-based saturation parameters in regular graphs.

References

If $G$ is an $r$-regular graph $G$ with $r > 0$, then $i(G) \le \mu*(G)$, and this bound is sharp. Moreover, the statement is clearly true for 2-regular graphs, and so, the conjecture is only open for $r$-regular graphs with $r \geq 3$.

— In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator (2507.17780 - Davila et al., 23 Jul 2025) in Section "Open conjectures of TxGraffiti (2016–2025)", Subsection "A mirror conjecture on independent domination and maximal matchings", Conjecture [TxGraffiti -- Open Since 2020]