The quarter-order bound for power domination in connected hypergraphs

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Let H=(V(H),E(H))\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H})) be a connected hypergraph on at least four vertices, with every edge eE(H)e\in E(\mathcal{H}) satisfying e3|e|\geq 3. Write γP(H)\gamma_{P}(\mathcal{H}) for its power domination number and γPI(H)\gamma_{P_{I}}(\mathcal{H}) for its infectious power domination number.

Quarter-order bound.

γPI(H)γP(H)V(H)4.\gamma_{P_{I}}(\mathcal{H})\leq\gamma_{P}(\mathcal{H})\leq\frac{|V(\mathcal{H})|}{4}.

The conjecture proposes that the observation or infection step improves the general domination bound from V(H)/3|V(\mathcal{H})|/3 to V(H)/4|V(\mathcal{H})|/4 for connected hypergraphs whose edges all have size at least three. Its status is not resolved in the supplied source context.

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Sources & referencesView supporting material

Primary source

Beth Bjorkman, “Infectious power domination of hypergraphs”, arXiv:1910.03038 (2019).

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