# List the cut vertices and bridges in the appropriate

This preview shows pages 2–4. Sign up to view the full content.

bridges, and blocks of G. List the cut-vertices and bridges in the appropriate places, and provide carefully labelled sketches of the blocks. Cut-vertices: Bridge(s): Block(s): _________________________________________________________________ 3. (10 pts.) Below, provide a proof by induction on the order of the graph G that every nontrivial connected graph G has a spanning tree. [Hint: If the order of the graph G is at least 3, Theorem 1.10 implies that G has a vertex v with G - v connected.]

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TEST2/MAD3305 Page 3 of 4 _________________________________________________________________ 5. (10 pts.) Apply Kruskal’s algorithm to find a minimum spanning tree in the weighted graph below. When you do this, list the edges in the order that you select them from left to right. What is the weight w(T) of your minimum spanning tree T? _________________________________________________________________ 6. (15 pts.) (a) Suppose G 1 and G 2 are nontrivial graphs. What does it mean mathematically to say that G 1 and G 2 are isomorphic?? [This is really a request for the definition!] (b) Sketch two graphs G and H that have the degree sequence s: 2, 2, 2, 2, 2, 2 and have the same order and size, but are not isomorphic. Explain briefly how one can readily see that the graphs are not isomorphic. (c) Explicitly realize C 5 and its complement below. [You may provide carefully labelled sketches.] Next, explicitly define an isomorphism from C 5 to its complement that reveals that C 5 is self-complementary.
TEST2/MAD3305 Page 4 of 4 _________________________________________________________________ 7. (10 pts.) Find a minimum spanning tree for the weighted graph below by using only Prim’s algorithm and starting with the vertex g. When you do this, list the edges in the order that you select them from left to right. What is the weight w(T) of your minimum spanning tree T? _________________________________________________________________ 8. (15 pts.) (a) If G is a nontrivial graph, how is κ (G), the vertex connectivity of G, defined? (b) If G is a nontrivial graph, it is not true generally that if v is an arbitrary vertex of G, then either κ (G - v) = κ (G) - 1 or κ (G - v) = κ (G). Give a simple example of a connected graph G illustrating this. [A carefully labelled drawing with a brief explanation will provide an appropriate answer.] (c) Despite the example above, if G is a nontrivial graph and v is a vertex of G, κ (G - v) κ (G) - 1. Provide the simple proof for this.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern