Homework 7

# Homework 7 - CS180 Winter 2011 Due 2nd March Homework 7...

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CS180 Winter 2011 Due: 2 nd March Homework 7 When submitting your homework, please include your name at the top of each page. If you submit multiple pages, please staple them together . We also ask that you do something to indicate which name is your last name on the first page, such as underlining it. Please submit your solutions with the problems solved in the order they are given. 1. In a school there are n boys and n girls. Each boy knows exactly k girls ( 1 k n ) and each girl knows exactly k boys. In this problem, “knowing” is mutual. (a) Prove that all the boys and girls can participate in one dance, where each pair of dancers (a boy and a girl) know each other. (b) Show that it is always true that k consecutive dances can be organized so that everyone will dance once with everyone he or she knows. 2. We define the Escape Problem 1 as follows. We are given a directed graph G = ( V, E ) (picture a network of roads). A certain collection of nodes X V are designated as populated nodes , and a certain other collection S V are designated as safe nodes . Assume that X and S are disjoint. In case of an emergency, we want evacuation routes from the populated nodes to

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