A circle criterion for a generalized cross graph in terms of minimal excluded minors

Abstract

Geelen and Oum described classes of minimal excluded pivot-minors for a simple graph to be a circle graph and for a delta-matroid to be Eulerian. Pivot-equivalence classes of circle simple graphs and delta-matroids arise in the investigation of Eulerian cycles on cross graphs (4-valent graphs with cross structure). The results established by Geelen and Oum rely on some lemmas in their work, which are shown below to be not quite correct. We consider generalized cross graphs, which arise in the description of rotating circuits on cross graphs. For such graphs we derive a circle criterion: we reproduce and augment the arguments due to Geelen and Oum, and we improve some incorrectly formulated statements. As a result, we obtain the same list of 166 inequivalent graphs, the minimal excluded minors for a generalized cross graph to be a circle graph. Bibliography: 14 titles.