3-dimensional Channel Routing

Consider two parallel planar grids of size w × n . The vertices of these grids are called terminals and pairwise disjoint subsets of termi nals are called nets. We aim at routing all nets in a cubic grid between the two layers h olding the terminals. However, to ensure solvability, it is allowed to introduce a n empty row/column be- tween every two consecutive rows/columns containing the te rminals (in both grids). Hence the routing is to be realized in a cubic grid of size 2 n × 2 w × h . The objective is to minimize the height h . In this paper we generalize previous results of Recski and Szeszl ́er [10] and show that every problem instance is so lvable in polynomial time with height h = O (max( n, w )). This linear bound is best possible (apart from a constant factor)

On the queue number of planar graphs

We prove that planar graphs have O(log2 n) queue number, thus improving upon the previous O(Formula Presented) upper bound. Consequently, planar graphs admit three-dimensional straight-line crossing-free grid drawings in O(n log8 n) volume, thus improving upon the previous O(n3/2) upper bound. © 2013 Society for Industrial and Applied Mathematics