# Graph minus a point has at most 2 connected components

Let $$G(V,E)$$ be a connected undirected graph (undirected=easier), such that, $$S_x=V\setminus\{x\}, \forall x \in V$$ is a subset of vertices of G. Then the induced subgraph $$G[S_x]$$ is the graph whose vertex set is $$S_x$$ and whose edges set consists of all of the edges in $$E$$ that have both endpoints in $$S_x$$.

Prove that a graph $$G(V,E)$$ has a Hamiltonian path (not necesarily a cycle) if the induced subgraphs $$\forall x \in V, G[S_x]$$ have at most 2 connected components.

• What is your thinking Dec 27, 2020 at 11:02
• What do you mean? Dec 27, 2020 at 11:03
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• My thoughs. math.stackexchange.com/questions/3963112/… Dec 27, 2020 at 11:07