{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

CO-342-1059-Midterm_exam

CO-342-1059-Midterm_exam - UNIVERSITY OF WATERLOO MIDTERM...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

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

View Full Document Right Arrow Icon
Background image of page 2
Background image of page 3

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

View Full Document Right Arrow Icon
Background image of page 4
Background image of page 5

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

View Full Document Right Arrow Icon
Background image of page 6
Background image of page 7

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

View Full Document Right Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UNIVERSITY OF WATERLOO MIDTERM EXAMINATION FALL TERM 2005 Surname: ________________.________ First Name: _______.______._.___ Id.#: _____________________.___ Course Number CO 342 Instructor J. Verstraete Date of Exam November 2, 2005 Duration: 2 hours Number of Exam Pages 8 (including this cover sheet) Exam Type Closed Book Additional Materials Allowed None Additional Instructions Calculators are not permitted. Answers are to be written clearly and legibly. Points may be deducted for unclear presentation. Problem Value Mark Awarded Value Mark Awarded 7 5 10 n 13 TOTAL 60 Question 1. A graph G is drawn in Figure 1. Let X be the set of vertices of G shown in grey. Figure 1 : The graph G ( ) On the drawing of G above, circle clearly the vertices in I‘(X). a (b) On the vertex set V1 below, draw G ~ X. (c) On the vertex set V2 below, draw G / X when X is contracted to :r. 0.1} (b) Vertex set V1 (c) Vertex set V2 0300M Question 2. (a) Define the concept of ear-decomposition for Q-edge-connected graphs, and state [4] the ear—decomposition theorem for 2—edge—connected graphs. (b) Let G be a 2-edge—connected graph in which every ear is a cycle. Prove that if G has exactly t ears, then G has at most if — 1 cut vertices. [Hint prove this by induction on t, starting with the case t = 1.] [3] (c) Draw an example of a 2—edge-connected graph in which every ear is a cycle [3] and in which there are t ears and exactly t —— 1 cut vertices. Question 3. State and prove the vertex—form of Menger’s Theorem. Define all concepts in [12] each of your statements. Question 4. (a) State Hall’s Theorem for a bipartite graph G (A, B) to have a perfect matching. [3] (b) Let G = G(A, B) be a three-regular bipartite graph. Prove that G — e has a [4] perfect matching for any edge e E E (G) Question 5. (a) State two conditions which together are necessary and sufficient for a bipartite [4] graph G’ (A, B) to have an f—factor. (b) Let C(A, B) be a bipartite graph such that every vertex of a has degree 2k, Where k 2 1, and every vertex of b has degree two. Prove that for any sets X C A and Y C B, e(X, Y) 2 2k1Xl— 213m. [Hint: use the equation e(X, Y) = e(X, B) — e(X7 B\Y).] [3] (0) Let C(A, B) be as in part (b), and let f : V(G) —> N be defined by f(a) = k [3] for a E A and f(b) = 1 for b E B. Using parts (a) and (b), prove that G(A, B) has an f—factor. Question 6. (a) Let C? be a directed graph, with a source-sink pair (5, t). Define the concepts [4] of an (s, t)—fiow and an (s, t)—cut in G. (b) Define the terms value of an (s,t)—flow and capacity of an (s,t)—cut, and then [3] state the Max—Flow Min—Cut theorem. (c) Consider the network C3 in Figure 2, where the capacities of the arcs of 63 are indicated by numbers next to each are (for example, the capacity of arc (b, a) is 2). Figure 2. 8 Let f be the (s, t)—flovv of value two in (3 defined by the following table: -_n- (i) Define an (37 t)—flow g in C? which augments the value of the flow f. When [3] defining this new flow, make a table of the non—zero values of the flow in each arc of G, as done for f above. (ii) List the arcs of a minimum (s,t)-cut of Ci, and explain why the flow g [3] found in (i) is a maximum (3, t)-flow in G. ...
View Full Document

{[ snackBarMessage ]}