On paths avoiding forbidden pairs of vertices in a graph

Abstract

Consider a digraph D = ( V, A) with two fixed vertices s, t ϵ V, and let F be a collection of mutually disjoint pairs of vertices throughout referred to as forbidden pairs. A directed ( s − t)-path is called F-path if it contains at most one vertex out of each pair in F. The problem F-PATH (FP) is: given D, s, t, and F, to decide if there exists an F-path in D. This problem, originated from the software testing and validation field (Krause, 1973), was shown by Gabow et al. (1976), to be NP-complete. In this paper the problem is studied under the following skew symmetry conditions: for each u, u′ and v, v′ in F, ( u, v) ϵ A implies ( v′, u′) ϵ A. The problem FP, when considered under the above conditions, is denoted throughout by SFP. SFP is shown here to have a polynomial solution. In particular, we show that SFP is polynomially equivalent to the well-known problem from the matching theory, of finding an augmenting path with respect to a given matching. Next, we present the dual F-cut notion, by means of which we establish a solvability criterion for SFP; namely: F-path exists in D if and only if D contains no F-cut.