6 research outputs found
Dominating sets in projective planes
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 is
smaller than (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 . 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.Comment: 19 page
On the stability of small blocking sets
A stability theorem says that a nearly extremal object can be obtained from an extremal one by âsmall changesâ. In this paper, we study the relation of sets having few 0-secants and blocking sets
On the metric dimension of affine planes, biaffine planes and generalized quadrangles
In this paper the metric dimension of (the incidence graphs of) particular
partial linear spaces is considered. We prove that the metric dimension of an
affine plane of order is and describe all resolving sets of
that size if . The metric dimension of a biaffine plane (also called
a flag-type elliptic semiplane) of order is shown to fall between
and , while for Desarguesian biaffine planes the lower bound is
improved to under , and to under certain
stronger restrictions on . We determine the metric dimension of generalized
quadrangles of order , arbitrary. We derive that the metric
dimension of generalized quadrangles of order , , is at least
, while for the classical generalized quadrangles
and it is at most