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Consider an undirected graph $G = [V,E]$. Let $V$ be the set of vertices: $V = \{v_1,..,v_n\}$ and $E$ be the set of edges. Let $C$ be the connected component that contains vertex $v_1$. I want to find the connected subgraph (not sure about this terminology) with maximum number of edges, among all the connected subgraph of $C$ that satisfy the following:

  1. includes vertex $v_1$
  2. has exactly $m$ number of vertices (including $v_1$)

By connected subgraph of $C$, I mean a subgraph of $C$, such that each pair of vertices in it are connected by a path.

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  • $\begingroup$ There is only one connected component containing a vertex. It doesn't seem like what you are saying is what you really want to ask. $\endgroup$ – plop Jul 14 at 20:34
  • $\begingroup$ What I meant is: find the subset (with max edges) of the connected component, among all the possible subset of size m of the connected component. I'm not sure about the exact terminologies. $\endgroup$ – Mamun Jul 14 at 20:40
  • $\begingroup$ @plop Please see if the edited question makes sense now. $\endgroup$ – Mamun Jul 14 at 21:04
  • $\begingroup$ @Mamun All above comments have become outdated since my edit. $\endgroup$ – John L. Jul 16 at 6:04
  • $\begingroup$ @D.W, can you remove all above comments? Thanks. $\endgroup$ – John L. Jul 16 at 6:04
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It's NP-hard (by reduction from clique problem).

Let $G$ be an arbitrary graph, and we want to check if there exists a clique of size $m-1$ in this graph. This problem is NP-hard. By adding $v_1$ to the graph and connecting it to all vertices in $G$, we reduce the clique problem to your problem (since clique, when exists, has the maximum number of edges).

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