@techreport{TR-IC-00-22,
  number = {IC-00-22},
  author = {Christiane N. Campos and Cláudio L. Lucchesi},
  title = {On the Relation between the {Petersen} Graph and the Characteristic of Separating Cuts of Matching Covered Graphs},
  month = {December},
  year = {2000}, 
  institution = {Institute of Computing, University of Campinas},
  note = {In English, 46 pages.
    \par\selectlanguage{english}\textbf{Abstract}
        A {\em matching covered graph} is a connected graph each edge
        of which lies in some perfect matching. A cut of a matching
        covered graph is {\em separating} if each of its two
        contractions yields a matching covered graph. A cut is {\em
        tight} if each perfect matching of the graph contains just one
        edge in the cut. Every tight cut of a matching covered graph
        is separating. The {\em characteristic} of a nontight
        separating cut is the smallest number of edges greater than
        one that some perfect matching of the graph has in the cut.
        The characteristic of a tight cut is defined to be equal to
        $\infty$.

        We show that the characteristic of every separating cut $C$ of
        a matching covered graph lies in $\{3,5,\infty\}$. Moreover,
        if $C$ has characteristic equal to 5 then graph $G$ has the
        Petersen graph as a minor, in a very strict sense. In
        particular, if $G$ is free of nontrivial tight cuts then $G$
        is the Petersen graph, up to multiple edges.
  }
}