General theoretical results on rectilinear embeddability of graphs.

*(English)*Zbl 0732.05021Summary: In the design of certain kinds of electronic circuits the following question arises: given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken line formed by horizontal and vertical segments and having at most k bends? Any such graph is said to be k-rectilinear. No matter what k is, an obvious necessary condition for k-rectilinearity is that the degree of each vertex does not exceed four.

Our main result is that every planar graph H 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 of H with at most 2 bends (3 bends if H is the octahedron) on each edge. The resulting embedding has the property that the total number of bends does not exceed 2n, where n is the number of vertices of H.

Our main result is that every planar graph H 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 of H with at most 2 bends (3 bends if H is the octahedron) on each edge. The resulting embedding has the property that the total number of bends does not exceed 2n, where n is the number of vertices of H.

##### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

94C15 | Applications of graph theory to circuits and networks |

##### Keywords:

rectilinear embeddability
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\textit{Y. Liu} et al., Acta Math. Appl. Sin., Engl. Ser. 7, No. 2, 187--192 (1991; Zbl 0732.05021)

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##### References:

[1] | Y. Liu, Boolean approach to planar embeddings of a graph,Acta Math. Sinica, New Series,5 (1989), 44–79. · Zbl 0780.05017 |

[2] | 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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