Roman Domination on Graphings

TL;DR

This paper investigates Roman domination in infinite measurable graphs called graphings, providing a framework for optimization and solving the problem on irrational cycle graphs.

Contribution

It introduces a framework for optimization in measurable graphings and solves the Roman domination problem on irrational cycle graphs.

Findings

Framework for optimization in measurable graphings

Complete solution for irrational cycle graphs

Advances understanding of domination in infinite graphs

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 ${\alpha}$.