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 Linear Layouts of Graphs

In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \pa

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