Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs

We study the properties of Schnyder’s realizers and canonical ordering trees of plane graphs. Based on these newly discovered properties, we obtain compact drawings of two styles for any plane graph G with n vertices. First we show that G has a visibility representation with height at most $\lceil \frac{15n}{16} \rceil$. This improves the previous best bound of n-1. The drawing can be obtained in linear time. Second, we show that every plane graph G has a straight-line grid embedding on an (n − Δ₀ − 1) × (n − Δ₀ − 1) grid, where Δ₀ is the number of cyclic faces of G with respect to its minimum realizer. This improves the previous best bound of (n-1) × (n-1). This embedding can also be found in O(n) time.