sol6 - CS 170 Algorithms Fall 2014 David Wagner Soln 6 1(20...

CS 170 Algorithms
Fall 2014 David Wagner
Soln 6
1. (20 pts.) Fundamental concepts
Decide whether each of the following claims is correct or not. If it is correct, write “True” and then provide
a short proof (one or two sentences should be enough). If it is incorrect, write “False” and provide a
counterexample (please make your counterexample as small as possible, for the readers’ sake).
(a) Let G be a dag and suppose the vertices v
1
,..., v
n
are in topologically sorted order. If there is an edge
( v
i
, v
j
) in G , then we are guaranteed that i < j .
(b) Let G be a dag and suppose the vertices v
1
,..., v
n
are in topologically sorted order. If there is a path
from v
i
to v
j
in G , then we are guaranteed that i < j .
(c) Let G = ( V , E ) be a directed graph. If there is an edge ( u , v ) in G , then u and v must be in the same
strongly connected component.
(d) Let G = ( V , E ) be a directed graph. If s ∈ V is a source and t ∈ V is a sink, then there is a path from s
to t .
CS 170, Fall 2014, Soln 6 1

(e) Let G = ( V , E ) be a directed graph where every vertex has at most three outgoing edges. Then every
vertex has at most three incoming edges.
(f) Let G = ( V , E ) be a dag. Then there are at most | V |
2
possible ways to order the vertices so they are in
topologically sorted order.