 
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
