Weak Positional Games on Hypergraphs of Rank Three
نویسنده
چکیده
In a weak positional game, two players, Maker and Breaker, alternately claim vertices of a hypergraph until either Maker wins by getting a complete edge or all vertices are taken without this happening, a Breaker win. For the class of almost-disjoint hypergraphs of rank three (edges with up to three vertices only and edge-intersections on at most one vertex) we show how to find optimal strategies in polynomial time. Our result is based on a new type of decomposition theorem which might lead to a better understanding of weak positional games in general.
منابع مشابه
The Angel Problem, Positional Games, and Digraph Roots
Preface This thesis is about combinatorial games—mostly. It is also about graphs, directed graphs and hypergraphs, to a large extent; and it deals with the complexity of certain computational problems from these two areas. We study three different problems that share several of the above aspects, yet, they form three individual subjects and so we treat them independently in three self-contained...
متن کاملExtremal Hypergraphs for the Biased Erdős-Selfridge Theorem
A positional game is essentially a generalization of Tic-Tac-Toe played on a hypergraph (V,F). A pivotal result in the study of positional games is the Erdős– Selfridge theorem, which gives a simple criterion for the existence of a Breaker’s winning strategy on a finite hypergraph F . It has been shown that the bound in the Erdős–Selfridge theorem can be tight and that numerous extremal hypergr...
متن کاملAvoider-Forcer Games on Hypergraphs with Small Rank
We study biased (a : f) Avoider-Forcer games played on hypergraphs, in the strict and monotone versions. We give a sufficient condition for Avoider’s win, useful in the case of games on hypergraphs whose rank is smaller than f . We apply this result to estimate the threshold bias in Avoider-Forcer (1 : f) games in which Avoider is trying not to build a copy of a fixed graph G in Kn. We also stu...
متن کاملBiased Positional Games and Small Hypergraphs with Large Covers
We prove that in the biased (1 : b) Hamiltonicity and k-connectivity MakerBreaker games (k > 0 is a constant), played on the edges of the complete graph Kn, Maker has a winning strategy for b ≤ (log 2− o(1))n/ log n. Also, in the biased (1 : b) Avoider-Enforcer game played on E(Kn), Enforcer can force Avoider to create a Hamilton cycle when b ≤ (1− o(1))n/ log n. These results are proved using ...
متن کاملUpper Bounds for Positional Paris-Harrington Games
We give upper bounds for a positional game — in the sense of Beck — based on the Paris-Harrington principle for bi-colorings of graphs and uniform hypergraphs of arbitrary dimension. The bounds show a striking difference with respect to the bounds of the combinatorial principle itself. Our results confirm a phenomenon already observed by Beck and others: the upper bounds for the game version of...
متن کامل