Dominating sets in projective planes

TL;DR

This paper investigates the structure of small dominating sets in the incidence graphs of finite projective planes, establishing stability results and characterizations that relate these sets to blocking and covering sets, especially in Desarguesian planes.

Contribution

It provides new bounds and structural characterizations of dominating sets in finite projective planes, linking them to blocking and covering sets, with specific results for planes of order greater than 81.

Findings

Dominating sets smaller than 2q+2[√q]+2 contain almost all points of a line or lines through a point.

Complete characterization of dominating sets of size at most 2q+√q+1.

In Desarguesian planes, small dominating sets are unions of blocking and covering sets, with few exceptions.

Abstract

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order $q>81$ is smaller than $2q+2[\sqrt{q}]+2$ (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most $2q+\sqrt{q}+1$. In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.