What is an example of two simple, undirected graphs $G,H$ such that
- there are no graph homomorphisms between $G, H$, and
- $H$ is not a minor of $G$, and $G$ is not a minor of $H$
?
Definition of minor: Let $G$ be a simple undirected graph. If $S, T$ are disjoint subsets of $V(G)$ we say that $S, T$ are connected to each other if there is $s\in S, t\in T$ such that $\{s,t\}\in E(G)$. If $G,H$ are graphs, we say $H$ is a minor of $G$ if there is a family ${\cal S}$ of pairwise disjoint connected subsets of $V(G)$ and a bijection $\varphi:V(H)\to {\cal S}$ such that whenever $\{v,w\}\in E(H)$ then $\varphi(v)$ and $\varphi(w)$ are connected to each other.
