Characterizing 3‐connected planar graphs and graphic matroids

Abstract

AbstractA well‐known result of Tutte states that a 3‐connected graph G is planar if and only if every edge of G is contained in exactly two induced non‐separating circuits. Bixby and Cunningham generalized Tutte's result to binary matroids. We generalize both of these results and give new characterizations of both 3‐connected planar graphs and 3‐connected graphic matroids. Our main result determines when a natural necessary condition for a binary matroid to be graphic is also sufficient. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 165–174, 2010