Extension and its price for the connected vertex cover problem

• Definition and study of the extension variants of the connected vertex cover problem ( Ext-CVC ), and of the Non-Separating Independent Set problem ( Ext-NSIS ). • Solvability and hardness results for Ext-CVC , notably in bipartite graphs of maximum degree 3 and in weakly triangulated graphs, applying also to Ext-NSIS . • Formulation of the concept of Price of Extension (PoE) as the approximation ratio for the problem Max Ext-CVC (resp., Min Ext-NSIS ). • Lower bounds for PoE of Max Ext-CVC in general and bipartite graphs; proof that it is equal to 1 in chordal graphs, carrying over to Min Ext-NSIS . We consider extension variants of Vertex Cover and Independent Set , following a line of research initiated in [10] . 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.