The Order Dimension of Planar Maps Revisited
نویسنده
چکیده
Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the Brightwell-Trotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyder-paths and Schnyder-regions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(PM)) 4. This may be the rst result in the area that is obtained without using the tools introduced by Schnyder. Mathematics Subject Classi cations (2010) 05C10, 06A07, 68R10
منابع مشابه
The Order Dimension of Planar Maps
This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3-connected planar map M, ordered by inclusion, and showed that the order dimension of PM is always equal to 4. In this paper, we show that if M i...
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عنوان ژورنال:
- SIAM J. Discrete Math.
دوره 28 شماره
صفحات -
تاریخ انتشار 2014