Abstract
A convex grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on grid points, all edges are drawn as straight-line segments between their endpoints without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of any 4-connected plane graph G with four or more vertices on the outer face boundary. The algorithm yields a drawing in an integer grid such that W + H ≤ n - 1 if G has n vertices, where W is the width and H is the height of the grid. Thus the area W x H of the grid is at most ⌈(n-1)=2⌉ · ⌈(n-1)/2⌉. Our bounds on the grid sizes are optimal in the sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n-1 and W x H = ⌈(n - 1)/2⌉ · ⌈(n - 1)/2⌉.
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Miura, K., Nakano, Si., Nishizeki, T. (2000). Convex Grid Drawings of Four-Connected Plane Graphs. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, SH. (eds) Algorithms and Computation. ISAAC 2000. Lecture Notes in Computer Science, vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40996-3_22
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DOI: https://doi.org/10.1007/3-540-40996-3_22
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