Generalized sandwich problem for Π - and ∆ - free multigraphs and its applications to positional extensions of game forms . a

An n-multigraph G = (V ;Ei | i ∈ I) is a complete graph G = (V,E) whose edges are covered by n = |I| sets E = ∪i∈IEi, some of which might be empty. If this cover is a partition, then we call G an n-graph. We say that an n-graph G′ = (V ;E′ i | i ∈ I) is an edge-subgraph of an n-multigraph G = (V ;Ei | i ∈ I) if E′ i ⊆ Ei for all i ∈ I. We denote by ∆ 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 Πand ∆-free n-subgraph, or not, and if yes provide a polynomial delay algorithm to generate all such subgraphs. The above decision problem can be viewed as a direct generalization of the sandwich problem for P4-free graphs introduced and solved by Golumbic, Kaplan, and Shamir in 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 Πand ∆-free n-graphs. Given 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.