Graphs with the same truncated cycle matroid

TL;DR

This paper extends Whitney's 2-Isomorphism Theorem to truncated cycle matroids, showing that most 3-connected graphs are uniquely identified by their truncated cycle matroid, except for K4.

Contribution

It introduces a characterization of graphs with the same truncated cycle matroid and proves uniqueness for all but K4 among 3-connected graphs.

Findings

Most 3-connected graphs are uniquely determined by their truncated cycle matroid.

The exception is the complete graph K4, which is not uniquely identified.

The paper generalizes classical matroid graph characterizations.

Abstract

The classical Whitney's 2-Isomorphism Theorem describes the families of graphs having the same cycle matroid. In this paper we describe the families of graphs having the same truncated cycle matroid and prove, in particular, that every 3-connected graph, except for K4, is uniquely defined by its truncated cycle matroid.