Let $V(\cdot), E(\cdot)$ denote the vertex and edge sets, respectively, of a given graph. Given two graphs $G$ and $H$, a (weak) homomorphism (definition 1.14) is a mapping $f:V(G) \to V(H)$ such that $(u,v) \in E(G) \implies (f(u),f(v))\in E(H) \lor f(u) = f(v)$.[1]
I am interested in the problem of find a weak homomorphism $f: V(G) \to V(H)$ that maximizes the cardinality of the image $|\{f(u) ; u \in V(G) \}|$. Alternatively, find a weak homomorphism $f: V(G) \to V(H)$ that maximizes the cardinality $|\{(s, t) \in E(H) ; \exists (u, v) \in E(G) \text{ such that } (s, t) = (f(u), f(v))\}|$.
Is there a name for such a problem?
Intuitively, I am interested the problem of glueing vertices of $G$ together to represent as much of $H$ as possible.
I found this for a start: maximum graph homomorphisms!