Abstract
We define a two-player combinatorial game in which players take alternate turns; each turn consists of deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player’s move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph’s automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games.
Highlights
In this article we define a two-person game played on the vertices of a graph and study it to find strategies for either player to win
We can create many different games from the same graph by giving each vertex a new random weight. As we studied these graphs, we discovered that we could replace a vertex with weight t with t “regular” vertices that do not share any edges
For multiple blowups we extend in the natural way the notation set of double blowups
Summary
In this article we define a two-person game played on the vertices of a graph and study it to find strategies for either player to win. Given a graph H, we define a legal move of Grim on H by a player selecting and deleting a vertex. When this vertex is deleted all edges adjacent to this vertex are deleted, together with any other vertices (if any) that have become isolated because of the move. The two players alternate turns, making legal moves on the follower that resulted from the previous player’s move They play until all vertices have been deleted.
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