Dominating Sets in Projective Planes

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⌊ √ 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 + √ 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.

Dominating sets in finite generalized quadrangles

“…The problem of domination has been studied before in incidence graphs of geometric structures, see for instance [6] and [11]. Also, perfect dominating sets of the incidence graphs of finite generalized quadrangles were considered in [4] (see also [9]), and for the particular quadrangle Q(4, q), they were studied in detail in [2]; see Section 5 for further information.…”

“…In W (q), or Q(4, q), if q is even, there exists an ovoid and a spread as well, giving rise to a dominating set of size 2q 2 + 2. This implies that there is no general stability phenomenon for smallest dominating sets in GQs, unlike in the case of generalized triangles (projective planes; see [11]); that is, the size of minimal examples (with respect to containment) may be arbitrarily close. However, the structure of the mentioned dominating sets are immensely dissimilar.…”

“…Perfect t fold dominating sets of the incidence graphs of projective planes, generalized quadrangles and generalized hexagons have already been studied in order to produce upper bounds on the order of some particular cage graphs (see [9] for an overview). In [5], perfect t-fold dominating sets of the incidence graph of the desarguesian projective plane PG(2, q) are completely described for small enough t, while the characterization of small dominating sets of projective planes can be found in [11]. In the case t = 1, describing smallest dominating and perfect dominating sets is quite easy, unlike in the here discussed case of generalized quadrangles; see also [2] for results on perfect (1-fold) dominating sets of Q(4, q).…”

Incidence‐free sets and edge domination in incidence graphs

“…It turns out that the problem of determining the domination number of I D ( ) is closely related to studying the smallest blocking sets and covers in D. Partially because of this, there have been several papers in recent years investigating the domination number of incidence graphs of block designs and other incidence structures. See, for example, [10,28] for incidence graphs of designs, [15] for incidence graphs of projective planes and [14] for incidence graphs of generalised quadrangles.…”