Vertex and edge covers with clustering properties: Complexity and algorithms

We consider the concepts of a t-total vertex cover and a t-total edge cover  (t⩾1)(t⩾1), which generalise the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t  , we present NPNP-completeness and approximability results (both upper and lower bounds) and FPTFPT algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t  -total edge cover and connected vertex cover, in particular improving on a previous FPTFPT algorithm for the latter problem.