Notes on a theorem of Naji

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.Comment: v1: 22 pages, no figure. v2: minor improvements. 23 pages, no figure v3: minor improvements. 23 pages, no figure v4: substantial improvements reflecting reader's suggestions. 32 pages, no figur

A Characterization of Circle Graphs in Terms of Multimatroid Representations

“…In order to do this we need a systematic way to choose oriented versions of touch-graphs. The approach we use is not the only possible one, but it is convenient because it is easy to describe and it is connected with signed interlacement systems that have been discussed by several authors [2,18,19,20,29,30].…”

“…Theorem 39. ( [21,22,29]) G is a circle graph if and only if the Naji equations of G have a solution over GF (2).…”

Naji's characterization of circle graphs

“…On the other hand, Geelen and Gerards gave a very short and intuitive proof of their characterization of graphic matroids. Motivated by that proof, Traldi gave a shorter proof of Naji's Theorem, but unlike the proof in , Traldi's proof is not self‐contained, relying on Bouchet's analogue of Tutte's Wheels Theorem for vertex‐minors, see . We give a self‐contained proof based on the methods presented in .…”

Bipartite complements of circle graphs

“…The known proofs of Naji's theorem are fairly involved [3,4,6,7], and Bouchet [1] (see also [2]) asked whether, on the other hand, Theorem 1 has an elementary proof. The purpose of this short note is to present such a proof.…”