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Unformatted text preview: Homework 5 Solutions Problem 1 9.4 Problem 55 Solution in text. 9.4 Problem 56 The graph is on the following page. Each node labeled ( x, y ) represents a state of the problem with x gallons in the 3-gallon jug and y gallons in the 5-gallon jug. In this problem the goal node can be of the form (1 , * ) or ( * , 1). There are two paths from the start node to goal nodes. Problem 2 (a) Each face of a square graph requires exactly one edge to triangulate it. Recall that the degree of a face is the number of edges on its boundary. In a square graph, each face has degree 4. Each edge appears on the boundary of the exactly 2 faces. Thus, if F is the set of faces, f the number of faces, and e the number of edges, then 2 e = X r ∈ F deg ( r ) = 4 f e = 2 f Since a square graph is also a planar graph, if n is the number of vertices, then from Euler’s formula 2 = n- e + f = n- 2 f + f Or, f = n- 2 Thus n- 2 edges are needed to triangulate a square graph with n vertices. 1 (0,0) (3,2) (0,2) (0,5) (3,5) (3,0) (1,5) (0,3) (3,3) (2,5) (3,4) (2,0) (0,4) (3,1) Start node in red, goal nodes in green....
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- Fall '07
- Graph Theory, Eulerian, 4 G, 5-gallon, square graph, 3-gallon