The Constructor-Blocker Game

TL;DR

This paper analyzes a combinatorial game involving two players claiming edges in a complete graph, aiming to maximize or minimize the number of certain subgraphs, and provides exact and asymptotic results for specific graph configurations.

Contribution

It determines the exact and asymptotic values of the game score for various pairs of fixed graphs, extending understanding of graph Turán-type games.

Findings

Exact value of g(n,H,F) for stars when both players play optimally.

Asymptotic behavior of g(n,H,F) for stars and trees.

Bounds on g(n,P_4,P_5) for the game configuration.

Abstract

We study the following game version of the generalized graph Tur\'an problem. For two fixed graphs $F$ and $H$, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph $K_n$. Constructor can only claim edges so that he never claims all edges of any copy of $F$, i.e. his graph must remain $F$-free, while Blocker can claim unclaimed edges without restrictions. The game ends when Constructor cannot claim further edges or when all edges have been claimed. The score of the game is the number of copies of $H$ with all edges claimed by Constructor. Constructor's aim is to maximize the score, while Blocker tries to keep the score as low as possible. We denote by $g(n,H,F)$ the score of the game when both players play optimally and Constructor starts the game. In this paper, we obtain the exact value of $g(n,H,F)$ when both $F$ and $H$ are stars and when $F=P_4$, $H=P_3$. We determine the asymptotics of $g(n,H,F)$ when $F$ is a star and $H$ is a tree and when $F=P_5$, $H=K_3$, and we derive upper and lower bounds on $g(n,P_4,P_5)$.