Minimal Blocking Sets Research Articles

Classification of minimal blocking sets in PG(2,9)

Classification of minimal blocking sets in PG(2,9)

Large blocking sets in PG(2,q2)

Minimal blocking sets in PG(2,q2) have size at most q3+1. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most q3+1−(p−3)/2, if q=p, p≥67, or q=ph, p>7, h>1. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).

Bridge-Depth Characterizes which Minor-Closed Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a predetermined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$ admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

Parameterized Complexity of Computing Maximum Minimal Blocking and Hitting Sets

A blocking set in a graph G is a subset of vertices that intersects every maximum independent set of G. Let \(\textsf {mmbs} (G)\) be the size of a maximum (inclusion-wise) minimal blocking set of G. This parameter has recently played an important role in the kernelization of Vertex Cover with structural parameterizations. We provide a panorama of the complexity of computing \(\textsf {mmbs} \) parameterized by the natural parameter and the independence number of the input graph. We also consider the closely related parameter \(\textsf {mmhs} \), which is the size of a maximum minimal hitting set of a hypergraph. Finally, we consider the problem of computing \(\textsf {mmbs} \) parameterized by treewidth, especially relevant in the context of kernelization. Since a blocking set intersects every maximum-sized independent set of a given graph and properties involving counting the sizes of arbitrarily large sets are typically non-expressible in monadic second-order logic, its tractability does not seem to follow from Courcelle’s theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem.

Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class $\mathcal{C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class $\mathcal{C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes $\mathcal{C}$, including the class $\mathcal{C}_{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or $\mathcal{C}_{LP}$ graphs.

Classification of minimal blocking sets in small Desarguesian projective planes

AbstractA full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.

On the stability of Baer subplanes

A blocking set in a projective plane is a point set intersecting each line. The smallest blocking sets are lines. The second smallest minimal blocking sets are Baer subplanes (subplanes of order q). Our aim is to study the stability of Baer subplanes in PG(2,q). If we delete q+1−k points from a Baer subplane, then the resulting set has size q+k and (q+1−k)(q−q) 0-secants. If we have somewhat more 0-secants, then our main theorem says that this point set can be obtained from a Baer subplane or from a line by deleting somewhat more than q+1−k points and adding some points. The motivation for this theorem comes from planes of square order, but our main result is valid also for non-square orders. Hence in this case the point set contains a relatively large collinear subset.

On the non-trivial minimal blocking sets in binary projective spaces

We prove that a non-trivial minimal blocking set with respect to hyperplanes in PG(r,2), r≥3, is a skeleton contained in some s-flat with odd s≥3. We also consider non-trivial minimal blocking sets with respect to lines and planes in PG(r,2), r≥3. Especially, we show that there are exactly two non-trivial minimal blocking sets with respect to lines and six non-trivial minimal blocking sets with respect to planes up to projective equivalence in PG(4,2). A characterization of an elliptic quadric in PG(5,2) as a special non-trivial minimal blocking set with respect to planes meeting every hyperplane in a non-trivial minimal blocking sets with respect to planes is also given.

On linear sets of minimum size

An Fq-linear set of rank k, k≤h, on a projective line PG(1,qh), containing at least one point of weight one, has size at least qk−1+1 (see De Beule and Van De Voorde (2019)). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between k∕2 and k−1. Our construction extends the known examples of linear sets of size qk−1+1 in PG(1,qh) constructed for k=h=4 (Bonoli and Polverino, 2005) and k=h in Lunardon and Polverino (2000). We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small k, we investigate whether all linear sets of size qk−1+1 arise from our construction.Finally, we modify our construction to define rank k linear sets of size qk−1+qk−2+…+qk−l+1 in PG(l,qh). This leads to new infinite families of small minimal blocking sets which are not of Rédei type.

Minimal Multiple Blocking Sets

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn - (3t + 1)(t - 1)} + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - \sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t \leq \sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $\mathrm{PG}(2,q)$ with $|S| = q\sqrt{q} + 1 + (t - 1)(q - \sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.

Blocking structures in finite projective planes

AbstractExploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 32h). Also, we show that these blocking sets of PG(2, q2), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 32h) gives rise to a family of blocking semiovals in the André plane as well.

On Bisecants of Rédei Type Blocking Sets and Applications

If B is a minimal blocking set of size less than 3(q+1)=2 in PG(2,q), q is a power of the prime p, then Szőnyi’s result states that each line meets B in 1 (mod p) points. It follows that B cannot have bisecants, i.e., lines meeting B in exactly two points. If q >13, then there is only one known minimal blocking set of size 3(q+1)=2 in PG(2, q), the so-called projective triangle. This blocking set is of Redei type and it has 3(q-1)=2 bisecants, which have a very strict structure. We use polynomial techniques to derive structural results on Redei type blocking sets from information on their bisecants. We apply our results to point sets of PG(2, q) with few odd-secants. In particular, we improve the lower bound of Balister, Bollobas, Furedi and Thompson on the number of odd-secants of a (q+2)-set in PG(2, q) and we answer a related open question of Vandendriessche. We prove structural results for semiovals and derive the non existence of semiovals of size q+3 when p≠3 and q>5. This extends a result of Blokhuis who classified semiovals of size q+2, and a result of Bartoli who classified semiovals of size q+3 when q ≤ 17. In the q even case we can say more applying a result of Szőnyi and Weiner about the stability of sets of even type. We also obtain a new proof to a result of Gacs and Weiner about (q+t, t)-arcs of type (0, 2, t) and to one part of a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi about functions over GF(q) determining less than (q+3)/2 directions.

Blocking and Double Blocking Sets in Finite Planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

Constructing Minimal Blocking Sets Using Field Reduction

AbstractWe present a construction for minimal blocking sets with respect to ‐spaces in , the ‐dimensional projective space over the finite field of order . The construction relies on the use of blocking cones in the field reduced representation of , extending the well‐known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino, and Storme (the MPS‐construction); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in F. Mazzocca and O. Polverino, J Algebraic Combin 24(1) (2006), 61–81. Furthermore, we show that every minimal blocking set with respect to the hyperplanes in can be obtained by applying field reduction to a minimal blocking set with respect to ‐spaces in . We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS‐constructionthen it is of Rédei‐type, whereas a small minimal blocking set arises from our cone construction if and only if it is linear.

Blocking sets of Hermitian generalized quadrangles

Some infinite families of minimal blocking sets on Hermitian generalized quadrangles are constructed.

Subregular spreads and blocking sets

We prove that any subregular spread in $$PG(3,q)$$ has an indicator set which defines a minimal blocking set in $$PG(2,q^2)$$ contained in three nonconcurrent lines (i.e., a rank three blocking set) and we discuss some examples.

Blocking Sets in

Abstract In this paper we provide a generalisation the MPS construction of a blocking set of P G ( r , q n ) using subspaces of dimension s ⩽ n − 2 , determining some minimal blocking sets in P G ( r , q 2 n )

A small minimal blocking set in PG(n, p t ), spanning a (t − 1)-space, is linear

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.

Minimal blocking set of size (30) in PG (2,19) plane

minimal blocking set of size (30) in PG (2,19) plane

Large weight code words in projective space codes

Recently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points.