I am assuming that the instructor is wanting us also to assume that graph G one has a cut vertex, then the compliment G' would have a different vertex as a cut vertex.
How would you find a graph G, such that both it and its complement would have cut vertices?
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graph-theory
2 Answers
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The graph $P_4$ (the path with four vertices) is isomorphic to its complement and clearly has a cut vertex.
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This construction generalizes Jorge Fernández Hidalgo's example.
Take a disconnected graph $G \cup H$ with $\geq 3$ vertices and no edges between $G$ and $H$. Add a vertex $v$, and connect it to all vertices in $G \cup H$ except one, $u$ say. By definition $v$ is a cut vertex of this graph.
An example is show below:
And its complement is:
In the complement, $v$ has the unique neighbor $u$, which has a neighbor other than $v$ (since $G \cup H$ is disconnected), highlighted in orange above. Thus $u$ is a cut vertex in the complement.