CSE 830: Design and Theory of Algorithms
Homework #4
Due Monday, March 17
th
2008, 3pm
1.
Is the path between a pair of vertices in a minimum spanning tree necessarily the shortest path
between the two vertices in the full graph? Give a proof or a counterexample.
2.
Suppose G is a connected undirected graph. An edge
e
whose removal disconnects the graph is
called a
bridge
. Must every bridge
e
be an edge in a depthfirst search tree of G, or can e be a back
edge? Give a proof or a counterexample.
3.
In breadthfirst and depthfirst search, an
undiscovered
node is marked
discovered
when it is first
encountered, and marked
completelyexplored
when it has been fully searched. At any given
moment, various numbers of nodes can be in any of these states. In all cases below, describe a
graph on n vertices with a particular starting vertex
v
that has the properties described.
a.
At some point during a breadthfirst search,
Θ
(
n
) are simultaneously in the discovered state.
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 Spring '08
 OFRIA
 Algorithms, Graph Theory, Glossary of graph theory, vertices, connected undirected graph, Theory of Algorithms

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