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We study the problem of finding the next-to-shortest paths in a graph. A next-to-shortest (u, v)-path is a shortest (u, v)-path amongst (u, v)-paths with length strictly greater than the length of the shortest (u, v)-path. In constrast to the situation in directed graphs, where the problem has been shown to be NP-hard, providing edges of length zero are(More)
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A graph G is called bisectable if it is an edge-disjoint union of two isomorphic subgraphs. We show that any tree T with e edges contains a bisectable subgraph with at least e-O(e/log log e) edges. We also show that every forest of size e, each component of which is a star, contains a bisectable subgraph of size at least e-O(log2 e).