TY - GEN
T1 - Convex drawings of internally triconnected plane graphs on O(n 2) grids
AU - Zhou, Xiao
AU - Nishizeki, Takao
N1 - Funding Information:
This work is supported in part by a Grant-in-Aid for Scientific Research (C) 19500001 from Japan Society for the Promotion of Science (JSPS).
PY - 2009
Y1 - 2009
N2 - In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n 3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n 3) to O(n 2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n×4n grid. We also present an algorithm to find such a drawing in linear time.
AB - In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n 3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n 3) to O(n 2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n×4n grid. We also present an algorithm to find such a drawing in linear time.
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U2 - 10.1007/978-3-642-10631-6_77
DO - 10.1007/978-3-642-10631-6_77
M3 - Conference contribution
AN - SCOPUS:75649145000
SN - 3642106307
SN - 9783642106309
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 760
EP - 770
BT - Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings
T2 - 20th International Symposium on Algorithms and Computation, ISAAC 2009
Y2 - 16 December 2009 through 18 December 2009
ER -