Journal of Combinatorial Optimization, 9, 35–47, 2005 c  2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Requiring Connectivity in the Set Covering Problem J. ORESTES CERDEIRA ∗ , † orestes@isa.utl.pt Instituto Superior de Agronomia, Univ. T´ ecnica de Lisboa, Tapada da Ajuda, 1349-017 Lisboa, Portugal LEONOR S. PINTO ‡ lspinto@iseg.utl.pt Instituto Superior de Economia e Gest ˜ ao, Univ. T´ ecnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal Received May 8, 2003; Revised July 26, 2004; Accepted September 21, 2004 Abstract. Given a bipartite graph with bipartition V and W , a cover is a subset C ⊆ V such that each node of W is adjacent to at least one node in C . The set covering problem seeks a minimum cardinality cover. Set covering has many practical applications. In the context of reserve selection for conservation of species, V is a set of candidate sites from a reserve network, W is the set of species to be protected, and the edges describe which species are represented in each site. Some covers however may assume spatial configurations which are not adequate for conservational purposes. Indeed, for sustainability reasons the fragmentation of existing natural habitats should be avoided, since this is recognized as being disruptive to the species adapted to the habitats. Thus, connectivity appears to be an important issue for protection of biological diversity. We therefore consider along with the bipartite graph, a graph G with node set V , describing the adjacencies of the elements of V , and we look for those covers C ⊆ V for which the subgraph of G induced by C is connected. We call such covers connected covers. In this paper we introduce and study some valid inequalities for the convex hull of the set of incidence vectors of connected covers. Keywords: set covering, graphs, connected components, integer polytopes MSC2000: 90C10, 90C57 1. Introduction Given a bipartite graph with bipartition V and W , a cover is a subset C ⊆ V such that each node of W is adjacent to at least one node in C . The set covering problem seeks a minimum cardinality cover. The problem has been extensively studied, and several papers have focused on the polyhedral structure of the convex hull of the set of incidence vectors of covers. Balas and Ng (1989a, b) characterize facets with coefficients and right hand sides in {0, 1, 2}.S´ anchez-Garc´ıa et al. (1998) extend this work to {0, 1, 2, 3}. Other classes of facets are study in Cornu´ ejols and Sassano (1989), Nobili and Sassano (1989), and Sassano (1989). Set covering has many practical applications (Vemuganti, 1998). It appears as a basic model in reserve selection for conservation of species. In this context, V is a set of candidate ∗ Corresponding author. † This author’s research was financially supported by the Portuguese Foundation for Science and Technology (FCT). ‡ This paper is part of this author’s Ph.D. research.