Unformatted text preview: AMS 303 , ‘ . TEST 2A Spring 2008 Prof. Tucker 1. Give the cycle structure representation for a 135 degree rotation of the corners of an 8gon. 2. a) Construct the matching network and make a ﬂow
corresponding to the partial matching c— i, d— —g, ej, fh. ' b) Apply the Augmenting Flow Algorithm (show
ALL labels) and from it obtain a 5—edge matching. 3. COnsider the game of Nim on the right. a) i 'I I I a) What is the Grundy number of the initial positon  I I I b) Make a move to get into a kernel position using pile 2. I I I I I
c) Make a move to get into a position with Grundy number equal to 1 I I I I I I I d) Suppose that 1 or 4 or 5 sticks "can be removed at a_time, determine is the Gurndy number of the initial
condition and then make a move into the kernel. 4. Give an expression for the'pattern inventory of
2colorings of the edges of the unoriented figure ‘
on the right (rotations and ﬂips allowed). 5. I. In the following table of remaining games, it is possible for the Bears to be cochampions (or outright champions) if they win all remaining games? Build the appropriate network model. Answer the questi0n with a feasible flow 1n the network you created or with an explanation of why one is not possible. If there are co
champions with the Bears, who are they? Wins Games with with with with Team to date to play Bears Lions Tigers Vampires
Bears ,20 6 — _ 1 2 ' '3
Lions 25 6 1 — 3 2
Tigers 25 6 2 3 — l
Vampires 23 6 3 . 2 1 — 11. Let G be a progressively finite graph with a kernel K and a Grundy function g( ). Form the graph G‘ by
deleting the vertices in the kernel K. Let K1 be the kernel of G'. Show that K1 is precisely the set of vertice with g(x)—  1 1n the original graph G (that IS, show that the vertices with g(x) — 1 in the original graph have
properties of a kernel 1n G). ' _ ...
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 Spring '08
 Tucker,A
 Graph Theory

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