Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 503 — #511
12.10. References 503
in the set are connected, and any set of vertices properly containing M will contain
two vertices that are not connected.
(a) What are the maximal connected subsets of the following (unconnected) graph?
(b) Explain the connection between maximal connected sets and connected components.
Prove it.
Problem 12.39. (a) Prove that K n is .n 1/-edge connected for n > 1.
Let M n be a graph defined as follows: begin by taking n graphs with nonoverlapping
sets of vertices, where each of the n graphs is .n 1/-edge connected
(they could be disjoint copies of K n , for example). These will be subgraphs of M n .
Then pick n vertices, one from each subgraph, and add enough edges between pairs
of picked vertices that the subgraph of the n picked vertices is also .n 1/-edge
connected.
(b) Draw a picture of M 3 .: : : M 4 /.
(c) Explain why M n is .n
1/-edge connected.
Problem 12.40.
False Claim. If every vertex in a graph has positive degree, then the graph is
connected.
(a) Prove that this Claim is indeed false by providing a counterexample.