I am a newbie to graph theory. If this is not the right forum to post this question on Graph theory please suggest the right forum.
I have a directed graph G with nodes 1, 2, 3,4. The adjacency matrix of G is
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 0
I would like to redraw/modify G by adding a relay/ extra node (Node: 5) such that the edges in G are still maintained but are now routed/channelled through the relay/extra node. That is, the nodes in the new graph are now: 1,2,3,4, 5 and the adjacency matrix of the new graph might look like:
0 1 0 0 2
0 0 0 0 2
0 0 0 1 0
0 0 0 0 0
0 0 2 2 0
I am a newbie to graph theory, so I am not sure if this is the correct way of rewriting a graph. If yes, how to check that the original graph's links/connections are still maintained when using Matlab?
As I understood, the modified graph $G’$ is consructed from $G$ such that any edge $(v,u)$ of $G$ is replaced by pair of edges $(v,5)$ and $(5,u)$. Since $(v,u)$ is a edge of $G$ iff $v<u$, we see that $G’$ have exactly edges $(v,5)$ and $5,u$, where $1\le v\le 3$ and $2\le u\le 4$. Then the adjacency matrix of $G’$ is