We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph G = (V, E) and a vertex set U ⊆ V , it is asked if there exists a minimal vertex cover (resp. maximal independent set) S with U ⊆ S (resp. U ⊇ S). Possibly contradicting intuition, these problems tend to be NP-hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of theses problems, with parameter |U |, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances.We further discuss the price of extension, measuring the distance of U to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.
We consider extension variants of Vertex Cover and Independent Set, following a line of research initiated in [9]. In particular, we study the Ext-CVC and the Ext-NSIS problems: given a graph G = (V, E) and a vertex set U ⊆ V , does there exist a minimal connected vertex cover (respectively, a maximal non-separating independent set) S, such that U ⊆ S (respectively, U ⊇ S). We present hardness results for both problems, for certain graph classes such as bipartite, chordal and weakly chordal. To this end we exploit the relation of Ext-CVC to Ext-VC, that is, to the extension variant of Vertex Cover. We also study the Price of Extension (PoE), a measure that reflects the distance of a vertex set U to its maximum efficiently computable subset that is extensible to a minimal connected vertex cover, and provide negative and positive results for PoE in general and special graphs.
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