A splitter for graphs with no Petersen family minor (Q1386424)

The Petersen family consists of the seven graphs that can be obtained from the Petersen graph by \(Y\Delta\)- and \(\Delta Y\)-exchanges. A splitter for a family of graphs is a maximal 3-connected graph in the family. In this paper, a previously studied graph, \(Q_{13,3}\), is shown to be a splitter for the class of all graphs with no Petersen family minor. Moreover, \(Q_{13,3}\) is a splitter for the family of graphs with no \(K_6\)-minor, as well as for the family of graphs with no Petersen minor.