Is there a cardinal $\kappa$ and a connected simple, undirected graph $G^\* = (\kappa, E^\*)$ such that whenever $G = (\kappa, E)$ is a connected graph, there is a [graph homomorphism](https://en.wikipedia.org/wiki/Graph_homomorphism) $f:G^\*\to G$?


**EDIT.** Sorry - I forgot the word "connected" in 2 positions of the question. Moreover, I changed the title according to [Peter Heinig](https://mathoverflow.net/users/108556/peter-heinig)'s comment (thanks, Peter!).



Connected weakly initial graph on $\kappa$ points
**Obsolete answer to a previous edit**: In the present statement the only possible $G^\*$ is the empty graph, for any non-empty $G^\*$ has no homomorphisms to empty $G$.


**After the edit (with additional requirement of connecitivity)**: for every $\kappa$, the initial objects $G^\*$ are exactly the connected bipartite graphs with vertex set $\kappa$. Indeed, parts of every bipartite $G^\*$ can be mapped to endpoints of any edge of $G$. On the other hand, if $G^\*$ is not bipartite, it does not have homomorphisms to bipartite $G$. The trivial special case is $\kappa \leq 1$, with the only object clearly being initial.


