# hw05 - u to v or from v to u or both u and v may lie on a...

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CS170 - Spring 2010 Problem Set 5 Out: Feb 28, 2010 Due: Mar 5, 2010, 4pm Please write your name, your student ID number, course name (CS170), homework number (this is HW#1), your TA’s name and your section number on the first page of all of your homeworks, and staple all the pages together. Please put your name, CS170 and homework number on all the pages (in case they get separated). Note DPV = Dasgupta, Papadimitriou, Vazirani refers to the textbook. So DPV 2.5 refers to Problem 5 in Chapter 2 of DPV. 1. DPV 4.1, 4.2, 4.4, 4.5, 4.13, 4.19, 4.21 2. A directed graph G = ( V, E ) is called semiconnected if for every pair of distinct vertices u and v there is either a path from
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Unformatted text preview: u to v or from v to u , or both ( u and v may lie on a cycle). Show that G is semiconnected if and only if the DAG formed by G ’s strongly connected components has a unique topologically sorted order, i.e. there is a unique way to order the DAG vertices v 1 , v 2 , . . . , v n such that any edge ( v i , v j ) satisﬁes i < j . Hint: Let ˜ G denote the DAG of strongly connected components of G . Show that G is semiconnected if and only if ˜ G has the following property: If v i and v i +1 are consecutive vertices in a topological sort of ˜ G , then the edge ( v i , v i +1 ) exists in ˜ G ....
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