An edge-ordered graph is an ordered pair (G, f), where G = G(V, E) is a graph and f is a bijective function, f : E(G) → {1, 2, ..., |E(G)|}. f is called an edge ordering of G. A monotone path of length k in (G, f) is a simple path P k+1 : v 1 , v 2 ..., v k+1 in G such that either, f ((v i , v i+1)) < f ((v i+1 , v i+2)) or f ((v i , v i+1)) > f ((v i+1 , v i+2)) for i = 1, 2, ..., k − 1. Given an undirected graph G, denote by α(G) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α(G) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.

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The results of Fujita and Farley (Discrete Appl. Math. 75 (1997) 255-268) and Kane and Peters (Discrete Appl. Math. 83 (1998) 207-228) on line broadcasting in paths and cycles are extended into a d-dimensional grid, obtaining optimal algorithms in most cases.

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Parallel elliptic single/multigrid solutions around an aligned and nonaligned body are presented and implemented on two multi-user and single-user shared memory multiprocessors (Sequent Symmetry and MOS) and on a distributed memory multiprocessor (a Transputer network). Our parallel implementation uses the Virtual Machine for Muli-Processors (VMMP), a software package that provides a coherent set of services for explicitly parallel application programs running on diverse multiple instruction multiple data (MIMD) multiprocessors, both shared memory and message passing. VMMP is intended to simplify parallel program writing and to promote portable and efficient programming. Furthermore, it ensures high portability of application programs by implementing the same services on all target multiprocessors. The performance of our algorithm is investigated in detail. It is seen to fit well the above architectures when the number of processors is less than the maximal number of grid points along the axes. In general, the efficiency in the nonaligned case is higher than in the aligned case. Alignment overhead is observed to be up to 200% in the shared-memory case and up to 65% in the message-passing case. We have demonstrated that when using VMMP, the portability of the algorithms is straightforward and efficient.