<XML><RECORDS><RECORD><REFERENCE_TYPE>3</REFERENCE_TYPE><REFNUM>8280</REFNUM><AUTHORS><AUTHOR>Fernau,H.</AUTHOR><AUTHOR>Manlove,D.F.</AUTHOR></AUTHORS><YEAR>2006</YEAR><TITLE>Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms</TITLE><PLACE_PUBLISHED>Proceedings of ACiD 2006: the 2nd Algorithms and Complexity in Durham workshop, volume 7 of Texts in Algorithmics, College Publications</PLACE_PUBLISHED><PUBLISHER>N/A</PUBLISHER><PAGES>69-84</PAGES><LABEL>Fernau:2006:8280</LABEL><KEYWORDS><KEYWORD>Algorithm; NP-completeness; approximability; t-total vertex cover; connected vertex cover; t-total edge cover</KEYWORD></KEYWORDS<ABSTRACT>We consider the concepts of a t-total vertex cover and a t-total edge cover (t>=1), which generalize 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 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-completeness and approximability results (both upper and lower bounds) and 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 algorithm for the latter problem.</ABSTRACT></RECORD></RECORDS></XML>