There are planar graphs almost as good as the complete graph
For a graph G represented in the plane (perhaps with edges intersecting internally) the length of a path joining two vertices of G is here regarded as the sum of the straight-line distances between successive vertices on the path. Given a set S of points in the plane, the complete graph on S has the property that for each pair A, B in S there exists an A-B path of length the straight-line distance between A and B. (The complete graph is unique with this property, if the points of S are in general position.) In this paper it is shown that there is a planar graph G on S with the property: for each pair A, B in S, there exists an A-B path of length at most twice the straight-line distance between A and B. The graph G is a type of Delaunay triangulation. It has 0 (ISI) edges (compare 0 \((ISI^ 2)\) for the complete graph). Applications are discussed to network design and motion planning.
scientific article; zbMATH DE number 4116563