König-Egerváry graphs, 2-bicritical graphs and fractional matchings

Abstract

A fractional node cover of a graph is an assignment of the values 0, $\text{1}\text{2}$, 1 to the nodes, so that for each edge, the sum of the values assigned to its two ends is at least one. Such a cover is minimum if the sum of the assigned values is minimized. A König-Egerváry graph is a graph for which there exists a minimum fractional cover in which all nodes receive the value 0 or 1. A 2-bicritical graph is one for which the unique minimum fractional cover is obtained by assigning $\text{1}\text{2}$ to all the nodes. We describe a polynomial method for decomposing a graph into 2-bicritical components and König-Egerváry components. This decomposition yields a minimum fractional node cover in which the number of nodes receiving the value $\text{1}\text{2}$ is minimized. We also show how excluded minor characterizations by Deming, Sterboul and Lovász of König-Egerváry graphs can be used to obtain a structural characterization of 2-bicritical graphs.