Splitting cubic circle graphs

TL;DR

This paper proves that all 3-regular circle graphs have at least two pairs of twin vertices, and identifies K_4 and K_{3,3} as the only 3-connected, 3-regular circle graphs up to isomorphism.

Contribution

It establishes structural properties of 3-regular circle graphs and characterizes the only 3-connected, 3-regular circle graphs.

Findings

Every 3-regular circle graph has at least two pairs of twin vertices.

No 3-regular circle graph is prime with respect to split decomposition.

K_4 and K_{3,3} are the only 3-connected, 3-regular circle graphs.

Abstract

We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected, 3-regular circle graphs.