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arXiv · 1603.02933

Dominating sets in projective planes

Tamás Héger·Zoltán Lóránt Nagy
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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.

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Tamás Héger, Zoltán Lóránt Nagy. 2016-03-09. Dominating sets in projective planes. https://doi.org/10.1002/jcd.21527

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