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CO-342-1059-Midterm_exam

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

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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. ...
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