Given any graph G, any pair of vertices (u.v), there exists a path length $d(u,v) \leq 3$ in G or there exists a path length $d(u,v) \leq 2$ in G'
graph-theory, every pair (u,v) of vertices has path length $\leq 3$ in G or $\leq 2$ in G'
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graph-theory
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0Riemann -- not much of any use, just fooling around with path lengths 4 or more and what that requires of G'. Rahu came up with a solution. I am burned out. – 2017-02-02
1 Answers
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Fix u,v
If there is a path of length less than 3 we are done.
Suppose that is not the case.So,given u and v there is no path of length less than 3.In particular G DOES NOT contain the edge (u,v)
So,the complement graph must contain the edge (u,v) whose length is $\leq 2$
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0Thank you Rahu. Question from a student and I am burned out. – 2017-02-02