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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−1)× (n−1) grid if either 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. An internally triconnected plane graph G has a convex grid drawing on a 2n × 2n grid if T (G) has exactly four leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 6n × n2 grid if T (G) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of polynomial size. key words: algorithm, convex grid drawing, graph drawing, plane graph, triconnected

Convex Grid Drawings of Plane Graphs with Pentagonal Contours