definition of flats

are flats always connected edges (it looks that way but I don’t see it as part of the definition)

are loops edges? flats?(1,1)

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3 comments

  1. I don’t think I understand the first question. Flats aren’t usually single edges – they are sets of edges.
    A loop is usually not a flat. In fact, if you think about it for a while, you will convince yourselves that any flat of a graph must contain all loops. (why?)

    • “A loop is not usually a flat” / “Any flat of a graph must contain all loops.”

      Loops are a bit poorly defined here – do you mean cycles or, for example, the edge (1,1)? Are you using the same meaning of “loop” in both those statements?

  2. But am I right to think it is true that in a simple graph single edges are *always* flats because we need at least 3 edges to form a cycle?Above , Federico says usually so I’m still not sure. I can see single edges wouldn’t be the majority of flats usually, but they always would be flats

    I’m reading the definition of flats to mean a flat contains either all the edges of any particular cycle or leaves out at least 2 edges of that cycle.
    This is consistent with any loops of a graph being included in all flats of that graph.
    Also if a loop is the only loop of a graph, then it is a flat.

    If someone sees somewhere I’ve gone wrong I would appreciate the comment

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