On finding the rectangular duals of planar triangular graphs
SIAM Journal on Computing
Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems
Theoretical Computer Science
Rectangle-of-influence drawings of four-connected plane graphs: extended abstract
APVis '05 proceedings of the 2005 Asia-Pacific symposium on Information visualisation - Volume 45
Convex Drawings of 3-Connected Plane Graphs
Algorithmica
Planar Polyline Drawings via Graph Transformations
Algorithmica
Straight-line drawing of quadrangulations
GD'06 Proceedings of the 14th international conference on Graph drawing
Polynomial area bounds for MST embeddings of trees
Computational Geometry: Theory and Applications
Closed rectangle-of-influence drawings for irreducible triangulations
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
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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.