On Open Rectangle-of-Influence Drawings of Planar Graphs

Abstract

We investigate open rectangle-of-influence drawings for irreducible triangulations, which are plane graphs with a quadrangular exterior face, triangular interior faces and no separating triangles. An open rectangle-of-influence drawing of a plane graph G is a type of straight-line grid drawing where there is no vertices drawn in the proper inside of the axis-parallel rectangle defined by the two end vertices of any edge. The algorithm presented by Miura and Nishizeki [8] uses a grid of size ${\cal W} + {\cal H} \leq$ (n-1), where ${\cal W}$ is the width of the grid, ${\cal H}$ is the height of the grid and n is the number of vertices in G. Thus the area of the grid is at most ⌈(n-1)/2⌉ × $\lfloor$(n-1)/2$\rfloor$ [8].

In this paper, we prove that the two straight-line grid drawing algorithms for irreducible triangulations from [4] and quadrangulations from [3,5] actually produce open rectangle-of-influence drawings for them respectively. Therefore, the straight-line grid drawing size bounds from [3,4,5] also hold for the open rectangle-of-influence drawings. For irreducible triangulations, the new asymptotical grid size bound is $\emph{11n/27}$ × $\emph{11n/27}$. For quadrangulations, our asymptotical grid size bound $\emph{13n/27}$ × $\emph{13n/27}$ is the first known such bound.