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Abstract. How small can a graph be that contains as subgraphs all trees on n vertices with maximum degree d? In this paper, this question is answered by constructing such universal graphs that have n vertices and bounded degree (depending only on d). Universal graphs with n vertices and O(n log n) edges are also constructed that contain all bounded-degree planar graphs on n vertices as subgraphs. In general, it is shown that the minimum universal graph containing all bounded-degree graphs on n vertices with separators of size n has O(n) edges if a < 1/2; O(n log n) edges if a 1/2; O(n2.) edges if a > 1/2.

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Ab8tract-Universal networks offer the advantage that they can execute programs written for simpler architectures without significant run-time overhead. In this paper we investigate simulations of tree machines; the fact that divideand-conquer algorithms are programmed naturally on trees motivates our investigation.Among various proposals for parallel computing the boolean hypercube has emerged as a particularly versatile network. It is well known that programs for multidimensional grid machines, for example, can be executed on a hypercube with no communications overhead by embedding the grid as a subgraph of the hypercube. Our first result is that a program for any tree machine can be executed on the hypercube with constant overhead. More precisely, every cycle of a synchronous binary tree can be simulated in 0(1) cycles on a hypercube, independent of the shape of the tree. The algorithm to embed the tree within the hypercube runs in polynomial time. We also give efficient simulations of arbitrary binary trees on the complete binary tree, the FFT and shufHe-exchange networks.It is natural to ask if any sparse network can simulate every binary tree efficiently. Somewhat surprisingly, we construct a universal bounded-degree network on N nodes for which every N node binary tree is a spanning tree. In other words, every binary tree can be simulated on our universal network with no overhead. This improves previous bounds on the sizes of universal graphs for trees.