A WellBehaved Extension of the Vertex Covering Problem Summary: A Well­Behaved Extension of the Vertex Covering Problem A. A. Ageev \Lambda Abstract The paper shows that several well­known results on properties of optimal solu­ tions of the minimum weight vertex covering problem (Balinski [1], Balinski and Spielberg [2], Nemhauser and Trotter [8], Hammer et al.[7], Bourjolly et al.[3]) remain true for an extension of it. 1 Introduction Let G = (V; E) be a undirected graph with vertex set V and edge set E. A subset X ` V is called a vertex covering if every edge of G has at least one endpoint in X. The minimum weight vertex covering problem (VCP) is, given a graph G with positive vertex weights c i (i 2 V ), to find a vertex covering X \Lambda with the minimum weight P i2X \Lambda c i . Clearly, a set X ` V is a vertex covering if and only if its complement V n X consists of pairwise nonadjacent vertices, i.e., is stable. So finding a minimum weight vertex covering is equivalent to finding a maximum weight stable set. Both problems are classic in discrete optimization and have been extensively investigated for recent decades. Despite NP­ hardness (see Garey and Johnson [5]) they enjoy a number of remarkable properties Collections: Mathematics