Is there a graph that is 1-connected and 3-connected, but not 4-connected? Why?
First off, can someone give the definition of being 1,3,or 4 connected in 'English'. All the def. are so technical and I don't really understand.
Intuitively a graph $\displaystyle G=\left(V,E\right)$ is, say $\displaystyle 2-connected$ (i.e. biconnected) if it has no, what is called "articulation points". These are points for which you can take away and leave the graph disconnected. In other words, no matter what vertex you take away from $\displaystyle G$ it remains connected. Think of this as a network flow having redundancies. If a server goes down there are other serves which still connect all the mainframes, and thus everything stays connected.