The novel octilinear routing paradigm (X-architecture) in VLSI design requires new approaches for the construction of Steiner trees. In this paper, we consider two versions of the shortest octilinear Steiner tree problem for a given point set K of terminals in the plane: (1) a version in the presence of hard octilinear obstacles, and (2) a version with rectangular soft obstacles. The interior o...

This paper develops techniques for computing the minimum weight Steiner triangulation of a planar point set. We call a Steiner point P a Steiner reducing point of a planar point set X if the weight (sum of edge lengths) of a minimum weight triangulation of X ∪{P} is less than that of X. We define the Steiner reducing set St(X) to be the collection of all Steiner reducing points of X. We provide...

Given a connected graph G = (V,E) (undirected, without loops and multiple edges) with positive edge costs (called also lengths) and a set Z ⊂ V of special (distinguished) vertices, the Steiner problem on graphs (networks) asks for a minimum cost tree within G that spans all members of Z. If |Z| = 2 we have the shortest path problem and if Z = V we get the minimum spanning tree problem, which ar...

We introduce bundle-free triangulations, that are free of large collection of triangles overlapping a circle empty of vertices. We prove that bundle-free Steiner triangulations can be used as an approximate solution for the minimum weight Steiner triangulation problem. We present new algorithms, implementations and experimental study for computing minimum weight Steiner triangulations.

The q-analogs of basic designs are discussed. It is proved that the existence of any unknown Steiner structures, the q-analogs of Steiner systems, implies the existence of unknown Steiner systems. Optimal q-analogs covering designs are presented. Some lower and upper bounds on the sizes of q-analogs covering designs are proved.

We consider the problem of finding a shortest rectilinear Steiner tree for a given set of points in the plane in the presence of rectilinear obstacles that must be avoided. We extend the Steiner ratio to the obstacle-avoiding case and show that it is equal to the Steiner ratio for the obstacle-free case.

Block-intersection graphs of Steiner triple systems are considered. We prove that the block-intersection graphs of non-isomorphic Steiner triple systems are themselves non-isomorphic. We also prove that each Steiner triple system of order at most 15 has a Hamilton decomposable block-intersection graph.

lichenized mycota of the southern part of Iran including Kerman province has been poorly studied compared to the Northern Iranian provinces such as Azerbaijan, Golestan or Mazandaran. Here we present the first lichen checklist for Kerman province, which comprises 57 species in 30 genera and 14 families. For this purpose, we reviewed both old and recent literature, examined herbarium collections...