Cycle Decompositions and Constructive Characterizations

TL;DR

This paper characterizes when Eulerian graphs have a unique cycle decomposition, providing a polynomial-time algorithm to decide this based on the absence of certain cycle-sharing structures.

Contribution

It offers a novel characterization of Eulerian graphs with unique cycle decompositions and introduces a polynomial-time decision algorithm.

Findings

Unique cycle decomposition occurs iff no two disjoint cycles share three or more vertices.

Introduces three binary graph operators for constructive characterizations.

Provides a polynomial-time algorithm for deciding cycle decomposition uniqueness.

Abstract

Decomposing an Eulerian graph into a minimum respectively maximum number of edge disjoint cycles is an NP-complete problem. We prove that an Eulerian graph decomposes into a unique number of cycles if and only if it does not contain two edge disjoint cycles sharing three or more vertices. To this end, we discuss the interplay of three binary graph operators leading to novel constructive characterizations of two subclasses of Eulerian graphs. This enables us to present a polynomial-time algorithm which decides whether the number of cycles in a cycle decomposition of a given Eulerian graph is unique.