Decomposition of a graph into two disjoint odd subgraphs

TL;DR

This paper characterizes when a multigraph can be partitioned into two odd subgraphs, providing a necessary and sufficient condition and a polynomial-time algorithm for such decompositions.

Contribution

It introduces a complete characterization and an efficient algorithm for decomposing multigraphs into two odd subgraphs, including the case involving even and odd subgraphs.

Findings

Provides a necessary and sufficient condition for the decomposition.

Develops a polynomial-time algorithm for the decomposition problem.

Addresses the case of decomposing into an even and an odd subgraph.

Abstract

An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both form odd subgraphs. In this paper we give a necessary and sufficient condition for the decomposability of a multigraph into two odd subgraphs. We also present a polynomial time algorithm for finding such a decomposition or showing its non-existence. We also deal with the case of the decomposability into an even subgraph and an odd subgraph.