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arXiv · 2404.19718

Roman Domination on Graphings

Adrian Rettich
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Abstract

We study a variant of domination, called Roman domination, where we must assign to each vertex one of the labels 0, 1, or 2 and require that every vertex with label 0 has a neighbour with label 2. We study the problem of finding a low-cost Roman dominating function on Lebesgue-measurable graphings, that is, on infinite graphs whose vertices are the points of a probability space. We provide a framework to tackle optimisation problems in the measurable combinatorial setting. In particular, we fully answer the Roman domination problem on irrational cycle graphs, a specific type of graphing on the space $\mathbb{R}/\mathbb{Z}$ where an irrational number ${\alpha}$ is given and two vertices are adjacent if and only if their distance is ${\alpha}$.

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Adrian Rettich. 2024-04-30. Roman Domination on Graphings. https://arxiv.org/abs/2404.19718

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