Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems
Theoretical Computer Science
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
New theoretical bounds of visibility representation of plane graphs
GD'04 Proceedings of the 12th international conference on Graph Drawing
Visibility representation of plane graphs via canonical ordering tree
Information Processing Letters
Visibility representations of four-connected plane graphs with near optimal heights
Computational Geometry: Theory and Applications
Visibility representation of plane graphs via canonical ordering tree
Information Processing Letters
Width-optimal visibility representations of plane graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Compact visibility representation of 4-connected plane graphs
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
An application of well-orderly trees in graph drawing
GD'05 Proceedings of the 13th international conference on Graph Drawing
Nearly optimal visibility representations of plane graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Compact visibility representation of 4-connected plane graphs
Theoretical Computer Science
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In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [Discrete Comput. Geom. 1 (1986) 343-353] and Tamassia and Tollis [Discrete Comput. Geom. 1 (1986) 321-341] independently gave linear time VR algorithms for 2-connected plane graph. Using this approach, the height of the VR is bounded by (n - 1), the width is bounded by (2n - 5). After that, some work has been done to find a more compact VR. Kant and He [Theoret. Comput. Sci. 172 (1997) 175-193] proved that a 4-connected plane graph has a VR with width bounded by (n - 1). Kant [Internat. J. Comput. Geom. Appl. 7 (1997) 197-210] reduced the width bound to ⌊3n-6/2⌋ for general plane graphs. Recently, using a sophisticated greedy algorithm, Lin et al. reduced the width bound to ⌊22n-42/15⌋ [Proc. STACS'03, Lecture Notes in Computer Science, vol. 2607, Springer, Berlin, 2003, pp. 14-25].In this paper, we prove that any plane graph G has a VR with width at most ⌊13n-24/9⌋, which can be constructed by using the simple standard VR algorithm in [P. Rosenstiehl, R.E. Tarjan, Discrete Comput. Geom. 1 (1986) 343-353; R. Tamassia, I.G. Tollis, Discrete Comput. Geom. 1 (1986) 321-341].