Sandwich problem for Π- and Δ-free multigraphs and its applications to positional games

An n-multigraph G = (V; E_i | i ϵ I) is a complete graph G = (V, E) whose edges are covered by n = |I| sets, E = ∪_iϵI E_i, some of which might be empty. If this cover is a partition, then G is called an n-graph. We say that an n-graph G′ = (V; E_i′ | i ϵ I) is an edge subgraph of an n-multigraph G = (V; E_i | i ϵ I) if E_i′ ⊆ E_i for all i ϵ I. We denote by Δ the n-graph on three vertices with three nonempty sets each containing a single edge, and by Π the four-vertex n-graph with two non-empty sets each of which contains the edges of a P₄. In this paper, we recognize in polynomial time whether a given n-multigraph G contains a Π- and Δ-free n-subgraph, or not, and if yes, provide a polynomial delay algorithm generating all such subgraphs. The above decision problem can be viewed as a generalization of the sandwich problem for P₄-free graphs solved by Golumbic et al. (1995). As a motivation and application, we consider the n-person positional game forms, which are known to be in a one-to-one correspondence with the Π- and Δ-free n-graphs. Given a game form g, making use of the above result, we recognize in polynomial time whether g is a subform of a positional (that is, tight and rectangular) game form and, if yes, we generate with polynomial delay all such positional extensions of g.