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Given an undirected graph $G$ (can be cyclic) with the promise that all its faces have $3$ sides, is it possible to find the minimum distance between a source and any other vertices in LogSpace or in LogDCFL, at least under the constraint that between any two vertices there are polynomially many paths?

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@StefanKohl: Please limit the flood of minor edits to a handful per day. See meta.mathoverflow.net/questions/599 . – Emil Jeřábek Oct 16 '15 at 15:42
    
@EmilJeřábek: o.k.. – Stefan Kohl Oct 16 '15 at 15:45
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What do you mean by "all its faces have 3 sides"? Faces are not a concept that makes sense for arbitrary (non-planar) graphs. And when you say "between any two vertices there are polynomially many paths", you mean simple paths, right? But in an undirected graph that's still a very strong restriction. – David Eppstein Oct 17 '15 at 4:43

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