The Largest Connected Subgraph Game

Abstract

This paper introduces the largest connected subgraph game played on an undirected graph G. In each round, Alice first colours an uncoloured vertex of G red, and then, Bob colours an uncoloured vertex of G blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than the order of any blue (red, resp.) connected subgraph. We first prove that, if Alice plays optimally, then Bob can never win, and define a large class of graphs (called reflection graphs) in which the game is a draw. We then show that determining the outcome of the game is PSPACE-complete, even in bipartite graphs of small diameter, and that recognising reflection graphs is GI-hard. We also prove that the game is a draw in paths if and only if the path is of even order or has at least 11 vertices, and that Alice wins in cycles if and only if the cycle is of odd length. Lastly, we give an algorithm to determine the outcome of the game in cographs in linear time.

The Maker-Breaker Largest Connected Subgraph game

“…Similarly as in [4], we establish some of our PSPACE-completeness results via reductions from POS CNF, a game for which deciding whether Alice or Bob has a winning strategy was shown to be PSPACE-complete in [22]. This game is a 2-player game where the input (X, φ) consists of a set of variables X = {x 1 , .…”

“…For both classes of graphs, we prove that c g can be determined in linear time. It is worth recalling that the outcome of the Largest Connected Subgraph game can be decided in linear time in cographs [4], while this was not proved for the more general class of (q, q − 4)-graphs.…”

“…The other motivation for introducing this game is that it is actually a Maker-Breaker version of the Largest Connected Subgraph game introduced in 2021 by Bensmail et al [4]. The Largest Connected Subgraph game opposes two players, Alice and Bob, through successive turns played on a graph G. Initially, every vertex of G is uncoloured.…”

“…Otherwise, they have the same score, and the game ends in a draw. In [4], it was proved that Alice always has a strategy to ensure at least a draw, and thus, that Bob can never win if Alice plays optimally. It was also proved that determining the outcome of the game on a given graph is PSPACE-complete, even when restricted to bipartite graphs of diameter 5, but polynomial-time solvable for paths, cycles, and cographs.…”

“…The Largest Connected Subgraph game was novel in that it is a very natural game, and there is also a rich background on these types of games. In particular, it is a connection game (see [7] for more on these games) since the players strive to create connected structures, and it is a scoring game (see [4] for more examples) since the winner is determined by the scores of the players. Thus, the Maker-Breaker Largest Connected Subgraph game is also a connection game.…”