Largest minimal blocking sets in PG(2,8)

Largest Minimal Blocking Sets in PG(2,8)

Bruen and Thas proved that the size of a large minimal blocking set is bounded by q ffiffiffi q p þ 1. Hence, if q 1⁄4 8, then the maximal possible size is 23. Since 8 is not a square, it was conjectured that a minimal blocking 23-set does not exist in PGð2; 8Þ. We show that this is not the case, and construct such a set. We prove that this is combinatorially unique. We also complete the spectr...

Minimal blocking sets in PG(2, 9)

We classify the minimal blocking sets of size 15 in PG(2, 9). We show that the only examples are the projective triangle and the sporadic example arising from the secants to the unique complete 6-arc in PG(2, 9). This classification was used to solve the open problem of the existence of maximal partial spreads of size 76 in PG(3, 9). No such maximal partial spreads exist [13]. In [14], also the...

On the largest minimal blocking set in PG(𝔽8)

Largest Minimal Percolating Sets in Hypercubes under 2-Bootstrap Percolation

Consider the following process, known as r-bootstrap percolation, on a graph G. Designate some initial infected set A and infect any vertex with at least r infected neighbors, continuing until no new vertices can be infected. We say A percolates if it eventually infects the entire graph. We say A is a minimal percolating set if A percolates, but no proper subset percolates. We compute the size ...

Largest and Smallest Minimal Percolating Sets in Trees

Bootstrap percolation is the process on a graph where, given an initial infected set, vertices with at least r infected neighbors are infected until no new vertices can be infected. A set percolates if it infects all the vertices of the graph, and a percolating set is minimal if no proper subset percolates. We consider the question for trees. We describe an O(n) algorithm for computing the larg...