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 NP NP -completeness and approximability results (both upper and lower bounds) and FPT FPT 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 FPT FPT algorithm for the latter problem.
t-total vertex cover t-total edge cover connected vertex cover fpt algorithm polynomial-time algorithm np-completeness approximation algorithm