Geodesic
Convex Grid Drawings of Plane Graphs with Rectangular Contours
Journal of Graph Algorithms and Applications, Tome 12 (2008) no. 2, pp. 197-224.

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In a convex drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n−1) ×(n−1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n ×n2 grid if T(G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. Our convex grid drawing has a rectangular contour, while most of the known algorithms produce grid drawings having triangular contours.
DOI : 10.7155/jgaa.00164
Keywords: graph drawing, convex grid drawing, plane graph, algorithm 
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Kazuyuki Miura; Akira Kamada; Takao Nishizeki. Convex Grid Drawings of Plane Graphs with Rectangular Contours. Journal of Graph Algorithms and Applications, Tome 12 (2008) no. 2, pp. 197-224. doi : 10.7155/jgaa.00164. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00164/

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