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
Proximity Drawability: a Survey
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
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
Planar open rectangle-of-influence drawings with non-aligned frames
GD'11 Proceedings of the 19th international conference on Graph Drawing
Open rectangle-of-influence drawings of non-triangulated planar graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
The approximate rectangle of influence drawability problem
GD'12 Proceedings of the 20th international conference on Graph Drawing
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A (weak) closed rectangle-of-influence (RI for short) drawing is a straight-line planar grid drawing in which there is no other vertex inside or on the boundary of the axis parallel rectangle defined by the two end vertices of any edge. Biedl et al. (1999) [1] showed that a plane graph G has a closed RI drawing, if and only if it has no filled 3-cycle (a cycle of 3 vertices such that there is a vertex in the proper interior). They also showed that such a graph G has a closed RI drawing in an (n-1)x(n-1) grid, where n is the number of vertices in G. They raised an open question on whether this grid size bound can be improved (Biedl et al., 1999 [1]). Without loss of generality, we investigate maximal plane graphs admitting closed RI drawings in this paper. They are plane graphs with a quadrangular exterior face, triangular interior faces and no filled 3-cycles, known as irreducible triangulations (Fusy, 2009 [2]). In this paper, we present a linear time algorithm that computes closed RI drawings for irreducible triangulations. Given an arbitrary irreducible triangulation G with n vertices, our algorithm produces a closed RI drawing with size at most (n-3)x(n-3); and for a random irreducible triangulation, the expected grid size of the drawing is (22n27+O(n))x(22n27+O(n)). We then prove that for arbitrary n=4, there is an n-vertex irreducible triangulation, such that any of its closed RI drawing requires a grid of size (n-3)x(n-3). Thus the grid size of the drawing produced by our algorithm is tight. This lower bound also answers the open question posed in Biedl et al. (1999) [1] negatively.