Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

Abstract

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

On the Metric Dimension of Imprimitive Distance-Regular Graphs

“…W. J. Martin (personal comunication) was able to construct a non-split resolving set for Γ Π of size 4q−4 (see [29, Figure 1]), and conjectured that this was best possible (except for small orders). This conjecture was also proved in the 2012 paper of Héger and Takáts [29]. Héger and Takáts also gave a complete description of all resolving sets of this size: see [29, §3].…”

“…There is computational evidence to support such a claim. It is known that strongly regular graphs with the same parameters need not have the same metric dimension: the Paley graph on 29 vertices has metric dimension 6, while the other strongly regular graphs with parameters (29,14,6,7), which fall into five switching classes, all have metric dimension 5 (see [5, Table 2]). Furthermore, the 3854 strongly regular graphs with parameters (35,16,6,8), which fall into exactly 227 switching classes [32], all have metric dimension 6 (see [5, Given what we know about the metric dimension of primitive strongly regular graphs from Theorem 1.2, we can combine this with Theorem 3.2 to obtain bounds on the metric dimension of Taylor graphs.…”

“…At the problem session of the 2011 British Combinatorial Conference, the author asked whether this was best possible. In 2012, the question was answered by Héger and Takáts [29] for the Desarguesian plane PG(2, q). (PG(2, q)) − 2}; for a square prime power q ≥ 121, this is at least q + √ q.…”

Blocking structures in finite projective planes

“…Recently, Dover [8] proved a lower bound on the size of a blocking semioval. See also [5], [11], for the Desarguesian case. Apart from the unitals there is a well-known example of blocking semioval, i.e.…”

Search problems in vector spaces

“…On the other hand, Bailey [1] gave a semi-resolving set of size τ 2 (PG(2, q)) −1, and Héger and Takáts [12] constructed one of size 2(q + √ q) in PG(2, q), q a square prime power. Recall that A(3, q) = 2q − 1 by Theorem 1.1.…”