Notes on a theorem of Naji

TL;DR

This paper offers a new algebraic proof of Naji's theorem, which characterizes circle graphs, and explores its implications for bipartite graphs, planar matroids, and permutation graphs.

Contribution

It provides a novel algebraic proof of Naji's theorem and connects it to characterizations of planar matroids and permutation graphs.

Findings

New algebraic proof of Naji's theorem

Equivalence of Naji's theorem to planar matroid characterization for bipartite graphs

Algebraic characterization of permutation graphs

Abstract

We present a new proof of an algebraic characterization of circle graphs due to W. Naji. For bipartite graphs, Naji's theorem is equivalent to an algebraic characterization of planar matroids due to J. Geelen and B. Gerards. Naji's theorem also yields an algebraic characterization of permutation graphs.