Let V be a finite set of points in the plane. In this paper, we show that the ratio of the lengths of Steiner minimal tree for V and the minimal spanning tree for V can never be less than

It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. We prove the conjecture for 3-connected cubic plane graphs and 3-connected cubic graphs on the projective plane. Our proof provides a polynomial time algorithm to find the decomposition for 3-connected cubic plane graphs.

Given a set S of n points in the plane, a minimumdilation spanning tree of S is a tree with vertex set S of smallest possible dilation. We show that given a set S of n points and a dilation δ > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most δ exists.

We introduce and study the orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of a plane graph H of G, and an orderly spanning tree of H. We also...

Because of the smooth interaction of tree systems, the billboard and crossed-plane techniques of image-based rendering (IBR) have been used for tree visualization for many years. However, both the billboard-based tree model (BBTM) and the crossed-plane tree model (CPTM) have several notable limitations; for example, they give an impression of slicing when viewed from the top side, and they prod...

Given n red and n blue points in the plane and a planar straight line matching between the red and the blue points, the matching can be extended into a bipartite planar straight line spanning tree. That is, any red-blue planar matching can be completed into a crossing-free red-blue spanning tree. Such a tree can be constructed in O(n log n) time. keywords: geometric graph, spanning tree, color

For point sets in the rectilinear plane we consider the following five measures of the interconnect length and prove bounds on the worst-case ratio: minimum Steiner tree, minimum star, clique, minimum spanning tree, and bounding box. In particular, we prove that for any set of n points: (n − 1) times the shortest Steiner tree is less or equal to the clique unless n = 4; and the minimum spanning...

This paper is concerned with applications of Tutte’s barycentric embedding theorem (1963, Proc. London Math. Soc. 13, 743–768). It presents a method for building isotopies of triangulations in the plane, based on Tutte’s theorem and the computation of equilibrium stresses of graphs by Maxwell–Cremona’s theorem; it also provides a counterexample showing that the analogue of Tutte’s theorem in di...