Abstract
In the design of certain kinds of electronic circuits the following question arises: given a non-negative integerk, what graphs admit of a plane embedding such that every edge is a broken line formed by horizontal and vertical segments and having at mostk bends? Any such graph is said to bek-rectilinear. No matter whatk is, an obvious necessary condition fork-rectilinearity is that the degree of each vertex does not exceed four.
Our main result is that every planar graphH satisfying this condition is 3-rectilinear: in fact, it is 2-rectilinear with the only exception of the octahedron. We also outline a polynomial-time algorithm which actually constructs a plane embedding ofH with at most 2 bends (3 bends ifH is the octahedron) on each edge. The resulting embedding has the property that the total number of bends does not exceed 2n, wheren is the number of vertices ofH.
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References
Y. Liu, Boolean approach to planar embeddings of a graph,Acta Math. Sinica, New Series,5 (1989), 44–79.
R. Tamassia and I. G. Tollis, A Provably Good Linear Algorithm for Embedding Graphs in the Rectilinear Grid, UILU-ENG-85-2233, ACT-64, Coordinated Science Laboratory, Uni. Illinois, 1985.
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Liu, Y., Morgana, A. & Simeone, B. General theoretical results on rectilinear embedability of graphs. Acta Mathematicae Applicatae Sinica 7, 187–192 (1991). https://doi.org/10.1007/BF02006104
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DOI: https://doi.org/10.1007/BF02006104