A study of convex grid drawing algorithms of plane graphs with contours of k-gon

在平面图形的凸图中，所有的边都绘制为没有任何边交的直线段，所有的面循环都绘制为凸多边形。在凸网格绘图中，所有顶点都放在网格点上。在本研究中，我们首先证明，如果三连通成分分解树T(G)恰好有四个叶子，则内部三连通平面图G在2n x 2n网格上绘制凸网格。我们还提出了一种在线性时间内找到这样的图的算法。然后，我们证明，如果T(G)恰好有五个叶子，则内部三连通的平面图G在6n x n2网格上绘制凸网格。我们还提出了一种在线性时间内找到这样的图的算法。此外，我们证明，如果T(G)恰好有六个叶子，则内部三连通的平面图G在6n x n2网格上绘制凸网格。我们还提出了一种在线性时间内找到这样的图的算法。

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.In this study, we first show that an internally triconnected plane graph G has a convex grid drawing on a 2n x 2n grid if the triconnected component decomposition tree T(G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. We then show that an internally triconnected plane graph G has a convex grid drawing on a 6n x n2 grid if T(G) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. Furthermore, We show that an internally triconnected plane graph G has a convex grid drawing on a 6n x n2 grid if T(G) has exactly six leaves. We also present an algorithm to find such a drawing in linear time.