Abstract
Let x 0,x 1,...x n-1 be vertices of a convex n-gon P in the plane, where, x 0 x 1, x 1 x 2, ... x n-2 x n-1, and x n-1 x 0 are edges of P. Let G = (N, E) be a graph, such that N = {0, 1, ..., n-1}. Consider a graph drawing of G such that each vertex i ∈ N is represented by x i and each edge (i, j) ∈ E is drawn by a straight line segment. Denote the sum of lengths of graph edges in such drawing by S P (G). If S P (G) ≤ S P (G′) for any convex n-gon P, then we write as G ≼ l G′. This paper shows two necessary and sufficient conditions of G ≼ l G′. Moreover, these conditions can be calculated in polynomial time for any given G and G’.
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References
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Ito, H., Uehara, H., and Yokoyama, M, A consideration on lengths of permutations on a vertex set of a convex polygon, Extended Abstracts of JCDCG’99, Tokai Univ., Nov. 26—27, pp. 40–41, 1999.
Ito, H., Uehara, H., and Yokoyama, M, Lengths of tours and permutations on a vertex set of a convex polygon, Discrete Applied Mathematics, (to appear).
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© 2001 Springer-Verlag Berlin Heidelberg
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Ito, H., Uehara, H., Yokoyama, M. (2001). Sum of Edge Lengths of a Graph Drawn on a Convex Polygon. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_14
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DOI: https://doi.org/10.1007/3-540-47738-1_14
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