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

An n -multigraph G = ( V ; E i ? i ? I ) is a complete graph G = ( V , E ) whose edges are covered by n = | I | sets, E = ? i ? I 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 ? I ) is an edge subgraph of an n -multigraph G = ( V ; E i ? i ? I ) if E i ' ? E i for all i ? I . We denote by Δ the n -graph on three vertices with three nonempty sets each containing a single edge, and by ? the four-vertex n -graph with two non-empty sets each of which contains the edges of a P 4 . In this paper, we recognize in polynomial time whether a given n -multigraph G contains a ? - 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 P 4 -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 ? - 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 .