Let $F$ and $G$ be graphs and $\varphi \colon F \to G$ a homomorphism. A vertex $a \in V(F)$ is $\varphi$-even if $|N_F(a) \cap \varphi^{-1}(x)|$ is even for every $x \in N_G(\varphi(a))$, and $\varphi$-odd if it is odd for every such $x$. The homomorphism $\varphi$ is an oddomorphism if

1. every vertex $a \in V(F)$ is $\varphi$-even or $\varphi$-odd, and
2. for every $x \in V(G)$, the fibre $\varphi^{-1}(x)$ contains an odd number of $\varphi$-odd vertices.

The homomorphism $\varphi$ is a weak oddomorphism if there is a subgraph $F' \subseteq F$ such that the restriction $\varphi|_{F'} \colon F' \to G$ is an oddomorphism. These notions are due to Roberson.

A class $\mathcal{F}$ of finite simple graphs is closed under weak oddomorphisms if $G \in \mathcal{F}$ whenever $F \in \mathcal{F}$ and there is a weak oddomorphism $F \to G$.