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**1 - 3**of**3**### Rectangle-of-Influence Drawings of Four-Connected Plane Graphs (Extended Abstract)

, 2005

"... A rectangle-of-influence drawing of a plane graph G is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of any edge. In this paper, weshow that any 4-connected plane graph G rectangle-of-influence drawing in an integer grid such that W + H n, where n is the numb ..."

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A rectangle-of-influence drawing of a plane graph G is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of any edge. In this paper, weshow that any 4-connected plane graph G rectangle-of-influence drawing in an integer grid such that W + H n, where n is the numberofvertices in G, W is the width and H is the height of the grid. Thus the area W \ThetaH of the grid is at most d(n;1)=2e\Delta b(n;1)=2c. Our bounds on the grid sizes are optimal in a sense that there exist an infinite number of 4connected plane graphs whose drawings need grids such that W +H = n;1andW \Theta H = d(n ; 1)=2e\Delta b(n ; 1)=2c. We also showthatthe drawing can be found in linear time.

### Convex Grid Drawings of Plane Graphs with . . .

, 2008

"... In a convex drawing of a plane graph, all edges are drawn as straightline 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 tric ..."

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In a convex drawing of a plane graph, all edges are drawn as straightline 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 â1) Ã(n â1) 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. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n Ã n 2 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.