Let $G=(V,E)$ be a graph. A 1-Lipschitz vertex projection is a map $p: E \to V$ such that $p(e)$ is always an end-vertex of $e$, and if $e,f$ have a common end-vertex, then $p(e)$ and $p(f)$ coincide or share an edge. The name is motivated by noticing that if we think of $G$ as an 1-complex, with each edge having length 1, and we think of $p$ as mapping midpoints of edges to $V$, then this map is 1-Lipschitz iff the aforementioned condition is satisfied.
Main Question Which graphs admit an 1-Lipschitz vertex projection?
It is easy to see that trees, cycles, and cliques do. Not every graph does (try e.g. piecing 4-cycles together).
Question 2: Does every chordal graph admit an 1-Lipschitz vertex projection?