The connected-graph visibility conjecture

A graph is representable as a visibility graph if it admits a visibility representation by regions whose unblocked sightlines correspond exactly to the edges of the graph. Connected-graph visibility conjecture. Every connected graph is representable as a visibility graph.

The conjecture is motivated by the construction showing that every connected planar graph can be realized as a visibility graph in the non-compact setting. The paper contrasts this with the fact that disconnected graphs such as two isolated vertices cannot be represented, so the connectedness hypothesis is essential.