Between Packable and Arbitrarily Packable Graphs: Packer-Spoiler Games

In a packer-spoiler game on a graph, two players jointly construct a maximal partial \(F\)-packing of the graph according to some rules, where \(F\) is some given graph. The packer wins if all the edges are used up and the spoiler wins otherwise. The question of which graphs are wins for which player generalizes the questions of which graphs are \(F\)-packable and which are randomly \(F\)-packable. While in general such games are NP-hard to solve, we provide partial results for \(F = P_3\) and solutions for \(F = 2K_2\).