If there exists a directed graph with $n$ vertices and $n$ edges, where each vertex has exactly one outgoing edge, and there exists multiple cycles (>=2), is it possible to form a graph where all the cycles are connected in one component?


closed as off-topic by Henrik, Misha Lavrov, jvdhooft, Chris Custer, A. Goodier Apr 11 '18 at 8:14

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  • $\begingroup$ A connected graph with $n$ vertices and $n-1$ edges is a tree. $\endgroup$ – saulspatz Apr 11 '18 at 5:31
  • $\begingroup$ Have you tried drawing such a graph to see if you can do it? $\endgroup$ – Misha Lavrov Apr 11 '18 at 6:19
  • $\begingroup$ @saulspatz notice it's $n$ edges and not $n-1$ edges $\endgroup$ – Simon Park Apr 11 '18 at 6:43
  • $\begingroup$ @Misha Lavrov I tried drawing many times but couldn't find an instance where two cycles could be in the same component, so I would like to believe it is not possible $\endgroup$ – Simon Park Apr 11 '18 at 6:44
  • $\begingroup$ @SimonPark Yes, you add one edge to a tree. How many cycle does that create? $\endgroup$ – saulspatz Apr 11 '18 at 14:44