Isotropic matroids. III: Connectivity

Robert Brijder¹ ; Lorenzo Traldi²

¹Hasselt University and Transnational University of Limburg
²Lafayette College

The electronic journal of combinatorics, Tome 24 (2017) no. 2

Résumé

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

Détail

DOI : 10.37236/5937

Classification : 05C40, 05B35, 52B40
Mots-clés : circle graph, connectivity, degree, isotropic system, local equivalence, matroid, pendant, prime, split, twin