# Example of an even subgraph that can't be decomposed into simple cycles?

In discussion of a lemma in the paper "Minimum cuts and shortest homologous cycles" Chambers Erickson and Nayyeeri make the claim that an even subgraph of an embedded graph can be decomposed into a union of weakly simple cycles but may not be able to be decomposed into a union of simple cycles. Does anyone know an example of an embedded graph that is like this?

Some definitions from the paper in case they are not standard:

A path in a surface $$\Sigma$$ is a continuous function $$p\colon [0,1]\to \Sigma$$. A path is simple if the function that defines it is injective. A cycle is a continuous function $$\gamma \colon S^1\to\Sigma$$.

An embedding of an undirected graph $$G$$ on a surface consists of a mapping of vertices of $$G$$ to distinct points in $$\Sigma$$ and a collection of mappings from edges of $$G$$ to simple paths in $$\Sigma$$ that intersect only at common endpoints.

Two paths in a combinatorial surface cross if no continuous infinitesimal perturbation makes them disjoint; if such a perturbation exists, then the paths are non-crossing. We say that a cycle $$\gamma$$ is non-self-crossing if no two sub-paths of $$\gamma$$ cross, weakly simple if $$\gamma$$ is non-self-crossing and traverses each edge at most once, and (strictly) simple if $$\gamma$$ visits each vertex at most once.

An even subgraph is a subgraph of $$G$$ in which every node has even degree, or equivalently, the union of edge-disjoint cycles

We define a cycle decomposition of an even subgraph $$H$$ to be a set of edge-disjoint, non-crossing, weakly simple cycles whose union is $$H$$.

Lemma 3.2. Every even subgraph of an embedded graph has a cycle decomposition.