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Supplementary information is available at http://www.cs.utoronto.ca/~juris/data/data/ppiGRG04/

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The graph isomorphism problem-to devise a good algorithm for determining if two graphs are isomorphic-is of considerable practical importance, and is also of theoretical interest due to its relationship to the concept of NP-completeness. No efficient (i.e.. polynomial-bound) algorithm for graph isomorphism is known, and it has been conjectured that no such algorithm can exist. Many papers on the subject have appeared, but progress has been slight; in fact, the intractable nature of the problem and the way that many graph theorists have been led to devote much time to it, recall those aspects of the four-color conjecture which prompted Harary to rechristen it the "four-color disease." This paper surveys the present state of the art of isomorphism testing, discusses its relationship to NP-completeness, and indicates some of the difficulties inherent in this particularly elusive and challenging problem. A comprehensive bibliography of papers relating to the graph isomorphism problem is given.

A tree t-spanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with m edges and n vertices is a minimum spanning tree and can be found in O(m log β(m, n)) time, where β(m, n) = min{i| log (i) n ≤ m/n}. On the other hand, for any fixed t > 1, the problem of determining the existence of a tree t-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree t-spanner is NP-complete for any fixed t ≥ 4. A theorem which captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an O((m + n)α(m, n)) algorithm is provided for finding a tree t-spanner with t as small as possible, where α(m, n) is a functional inverse of Ackerman's function. The results for tree spanners on undirected graphs are extended to "quasitree spanners" on digraphs. Furthermore, linear time algorithms are derived for verifying tree spanners and quasitree spanners.

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