Extension of some edge graph problems: Standard, parameterized and approximation complexity

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems and generalizations thereof. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (or maximal, respectively) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (or a matching, respectively) and containing the forced edge set U (or avoiding any edges from the forbidden edge set E∖U, respectively). 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 and inapproximability results.