Strong and weak edges of a graph and linkages with the vertex cover problem

Authors:
Qiaoming Han;Abraham P. Punnen
Affiliations:
School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, Zhejiang 310018, PR China;Department of Mathematics, Simon Fraser University, 13450 102nd Ave., Surrey, BC V3T5X3, Canada
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

In this paper we show that the problem of identifying an edge (i,j) of a graph G such that there exists an optimal vertex cover S of G containing exactly one of the vertices i and j is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee 2-11+@s, where @s is an upper bound on a measure related to a weak edge of a graph. A related problem of identifying an edge (i,j) such that there exists an optimal vertex cover containing both vertices i and j is also shown to be NP-hard. Further, we discuss a new relaxation of the vertex cover problem which is used in our approximation algorithm to obtain smaller values of @s. We also obtain linear programming representations of the vertex cover problem on special graphs.