Convex grid drawings of plane graphs with rectangular contours

  • Authors:

  • Akira Kamada;Kazuyuki Miura;Takao Nishizeki

  • Affiliations:

  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan;Faculty of Symbiotic Systems Science, Fukushima University, Fukushima, Japan;Graduate School of Information Sciences, Tohoku University, Sendai, Japan

  • Venue:
  • ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
  • Year:
  • 2006

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Abstract

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 ×n grid if 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.