Miloš Stojaković

Lucas Meijer, Till Miltzow, Johanna Ockenfels et al.

In the (Nesting) Bird Box Problem we are given a polygonal domain P and a number k and we want to know if there is a set B of k points inside P such that no two points in B can see each other. The underlying idea is that each point represents a birdhouse and many birds only use a birdhouse if there is no other occupied birdhouse in its vicinity. We say two points a,b see each other if the open segment ab intersects neither the exterior of P nor any vertex of P. We show that the Nesting Bird Box problem is ER-complete. The complexity class ER can be defined by the set of problems that are polynomial time equivalent to finding a solution to the equation $p(x) = 0$, with $x\in R^n$ and $p\in $Z[X_1,...,X_n]$. The proof builds on the techniques developed in the original ER-completeness proof of the Art Gallery problem. However our proof is significantly shorter for two reasons. First, we can use recently developed tools that were not available at the time. Second, we consider polygonal domains with holes instead of simple polygons.

Miloš Stojaković, Nikola Trkulja

We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge set of a complete graph $K_n$, where Maker's goal is to claim a Hamiltonian cycle. First, we prove that, independent of who starts, Maker can win the game for $n = 8$ and $n = 9$. Then we use an inductive argument to show that, independent of who starts, Maker can win the game if and only if $n \geq 8$. This, in particular, resolves in the affirmative the long-standing conjecture of Papaioannou. We also study two standard positional games related to Hamiltonicity game. For Hamiltonian Path game, we show that Maker can claim a Hamiltonian path if and only if $n \geq 5$, independent of who starts. Next, we look at Fixed Hamiltonian Path game, where the goal of Maker is to claim a Hamiltonian path between two predetermined vertices. We prove that if Maker starts the game, he wins if and only if $n \geq 7$, and if Breaker starts, Maker wins if and only if $n \geq 8$. Using this result, we are able to improve the previously best upper bound on the smallest number of edges a graph on $n$ vertices can have, knowing that Maker can win the Maker-Breaker Hamiltonicity game played on its edges. To resolve the outcomes of the mentioned games on small (finite) boards, we devise algorithms for efficiently searching game trees and then obtain our results with the help of a computer.