A note on minimal visibility graphs

Two vertices of a simple polygon $\text{P}$ are visible if the line segment joining them does not intersect the exterior of $\text{P}$. The visibility graph of $\text{P}$ is a graph $\text{G}$ obtained by representing each vertex of $\text{P}$ by a vertex of $\text{G}$ and two vertices of $\text{G}$ are joined by an edge if and only if their corresponding vertices in $\text{P}$ are visible. A graph $\text{G}$ is a visibility graph if there exists a simple polygon $\text{P}$ such that $\text{G}$ is isomorphic to the visibility graph of $\text{P}$. No characterization of visibility graphs is available. Coullard and Lubiw derived a necessary condition for a graph to be the visibility graph of a simple polygon. They proved that any 3-connected component of a visibility graph has a 3-clique ordering starting from any triangle. However, Coullard and Lubiw insisted on the non-standard definition of visibility (the line segment joining two visible vertices may not go through intermediate vertices) and they said they do not know if their result will hold under the standard definition of visibility. The purpose of this paper is to prove that Coullard and Lubiw's result still holds under the standard definition of visibility.

Two vertices $\text{A}$ and $\text{B}$ of a simple polygon $\text{P}$ are (mutually) visible if $\text{AB}$ does not intersect the exterior of $\text{P}$. A graph $\text{G}$ is a visibility graph if there exists a simple polygon $\text{P}$ such that each vertex of $\text{G}$ corresponds to a vertex of $\text{P}$ and two vertices of $\text{G}$ are joined by an edge if and only if their corresponding vertices in $\text{P}$ are visible. No characterization of visibility graphs is available. Abello, Lin and Pisupati conjectured that every hamiltonian maximal planar graph with a 3-clique ordering is a visibility graph. In this paper, we disprove this conjecture.

We prove the tight lower bound 4n − 4 on the size of tangent visibility graphs on n pairwise disjoint bounded obstacles in the euclidean plane, and we give a simple description of the configurations of convex obstacles which realize this lower bound.