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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
2
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Cut locus in a graph
Hopefully, I didn't completely misread the definitions. So here goes.
If we use definition 1. of distinct, (b) could be simplified to "all edges between vertices at the same distance from x" (and the …