Given a simple rectilinear polygon P with k sides and n terminals on its boundary, we present an O(k 3 n)-time algorithm to compute the minimal rectilinear Steiner tree lying inside P interconnecting the terminals. We obtain our result by proving structural properties of a selective set of minimal Steiner trees and exploiting them in a dynamic programming algorithm.

Given a directed graph G = (V,E) and an integer k ≥ 1, a Steiner k-transitive-closure-spanner (Steiner k-TC-spanner) of G is a directed graph H = (VH , EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V , the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we conce...

Two new approximate algorithms with O ( k l o g k ) for the rectilenear Steiner tree are proposed. Both algorithms base upon the method which makes minimum spanning tree on the modified Delaunay net with the triangular Steiner points as the more virtual generating points, because each point and each edge weight on the modified Delaunay net have been fixed.

A (k; g)-cage is a k-regular graph of girth g with minimum order. In this work, for all k ≥ 2 and g ≥ 5 odd, we present an upper bound of the order of a (k; g + 1)-cage, which depends on the order of a (k; g)-cage, improving a previous result of Sauer of 1967. We also show that every (k; 11)-cage contains a cycle of length 12, confirming a case of a conjecture of Harary and Kóvacs of 1983.

P. T. Araujo,1 I. O. Maciel,1 P. B. C. Pesce,1 M. A. Pimenta,1 S. K. Doorn,2 H. Qian,3 A. Hartschuh,3 M. Steiner,4 L. Grigorian,5 K. Hata,6 and A. Jorio1,7 1Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais 30123-970 Brazil 2Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3Departament Chemie und Biochemie and CeNS, L...

In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to ζ(3) = 1/13 + 1/23 + 1/33 + · · · as n → ∞. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k ≥ log2 log n+ω(1), where ω(1) is any function going to ∞ with n, then th...

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Stein...