A Note on Double Ideals: Connections to Width Parameters and Obstructions in Game Theory

Cops and Robbers is a well-known game in the fields of graph theory and game theory. In Cops and Robbers, one of the winning strategies for the cop (or the robber) is known as the (k,m)-obstacle. It is known that the (k,m)-obstacle has a dual relationship with graph width parameters. The concept of an "ideal" holds significance in topology and algebra. In this short paper, we propose a double ideal with the idea of element extension in the Matroid field, and show that it is equivalent to the (k,2)-obstacle. This paper offers little in terms of novelty, but I hope it will attract readers who are interested in game theory, graph width parameters, and algebraic studies.