Extension of Some Edge Graph Problems: Standard and Parameterized Complexity

Abstract

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems. Given a graph G=(V,E) and an edge set (formula presented), it is asked whether there exists an inclusion-wise minimal (resp., maximal) feasible solution E' which satisfies a given property, for instance, being an edge dominating set (resp., a matching) and containing the forced edge set U (resp., avoiding any edges from the forbidden edge set E\U). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation results.