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Is there a sub-field of graph theory that deals with weighted, directed graphs where two nodes are either connected in both directions, or neither?

A simple real life application would be a hilly park with many trails and you want to find the fastest way to traverse all trails.

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A directed graph with no one-way connections is just an undirected graph. –  you-sir-33433 Oct 9 '13 at 18:22
    
The weights are different from A->B than B->A. –  Balazs Rau Oct 9 '13 at 18:54
    
It is just special form of a weighted directed graph. –  Fraukje Oct 12 '13 at 20:09
    
@Fraukje: right, but are you aware of theorems that look at directed graph in a way that for example can solve the problem is finding an Eulerian path where is A->B is traversed then you don't need to traverse B->A ? Just like the question I asked about trails in a hilly park. –  Balazs Rau Oct 14 '13 at 16:53
    
@BalazsRau I am not that into specific theorems, but to the best of my knowledge there is no special term for graphs as you describe. –  Fraukje Oct 14 '13 at 19:16
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