The minimum feature size of a crossing-free straight line drawing is the minimum distance between a vertex and a non-incident edge. This quantity measures the resolution needed to display a figure or the tool size needed to mill the figure. The spread is the ratio of the diameter to the minimum feature size. While many algorithms (particularly in meshing) depend on the spread of the input, none...

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices a...

The Group Steiner Problem (GSP) is a generalized version of the well known Steiner Problem. For an undirected, connected distance graph with groups of required vertices and Steiner vertices, GSP asks for a shortest connected subgraph, containing at least one vertex of each group. As the Steiner Problem is NP-hard, GSP is too, and we are interested in approximation algorithms. EEcient approximat...

Let R be a finite set of terminals in a metric space (M,d). We consider finding a minimum size set S ⊆ M of additional points such that the unit-disc graph G[R ∪ S] of R ∪ S satisfies some connectivity properties. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem G[R ∪ S] should be connected. In the more general Steiner Forest with Minimum Number of Steiner Points (SF-M...

For a graph G and a positive integer k, the k-power of G is the graph G with V (G) as its vertex set and {(u, v)|u, v ∈ V (G), dG(u, v) ≤ k} as its edge set where dG(u, v) is the distance between u and v in graph G. The k-Steiner root problem on a graph G asks for a tree T with V (G) ⊆ V (T ) and G is the subgraph of T k induced by V (G). If such a tree T exists, we call it a k-Steiner root of ...

Given a weighted graph G = (V,E) and a subset R of V , a Steiner tree in G is a tree which spans all vertices in R. The vertices in V \R are called Steiner vertices. A full Steiner tree is a Steiner tree in which each vertex of R is a leaf. The full Steiner tree problem is to find a full Steiner tree with minimum weight. The bottleneck full Steiner tree problem is to find a full Steiner tree wh...

25 The Traveling Salesman Problem (TSP) is still one of the most researched topics in computational mathematics, and we introduce a variant of it, namely the study of the closed k-walks in graphs. We 2 G. Bullington, R. Gera, L. Eroh And S.J. Winters search for a shortest closed route visiting k cities in a non complete graph without weights. This motivates the following definition. Given a set...

Ting, S.-T. and S.-Y. Zhao, The general Steiner problem in Boolean space and application, Discrete Mathematics 90 (1991) 75-84. The Steiner problem is to find the minimum point P of sum of distances C;=‘=,p=@(P)whereP,i=l,..., n, are given points in Euclidean plane. It is solved for n = 3, [l]. When n > 3 even approximation is unsatisfactory [3]. In Boolean space the distance of Boolean points ...

This article considers a family of functionals J to be maximized over the planar convex sets K for which the perimeter and Steiner point have been fixed. Assuming that J is the integral of a quadratic expression in the support function h, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main the...