Abstract
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 [11], Tamassia and Tollis [14] independently gave linear time VR algorithms for 2-connected plane graph. Recently, Lin et. al. reduced the width bound to \(\lfloor \frac{22n - 42}{15} \rfloor\) [10]. In this paper, we prove that any plane graph G has a VR with width at most \(\lfloor \frac{13n - 24}{9} \rfloor\).
For a 4-connected plane triangulation G, we give a visibility representation of G with height at most \(\lceil \frac{3n}{4} \rceil\). In order to show that, we first show that every such graph has a canonical ordering tree with at most \(\lceil \frac{n+1}{2} \rceil\) leaves instead of the previously known bound \(\lfloor \frac{2n + 1}{3} \rfloor\), which is of independent interest. All of them can be obtained in linear time.
Research supported in part by NSF Grant CCR-0309953.
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Zhang, H., He, X. (2004). On Visibility Representation of Plane Graphs. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_42
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DOI: https://doi.org/10.1007/978-3-540-24749-4_42
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