CSE 8355
Graph Algorithms & Applications
Sample Quiz
D. W. Matula
I. [4 points each] TERMINOLOGY Give one or two sentence descriptions of each
of the following terms.
1.
Cycle: _____________________________________________________
________________________________________________________________
2.
Induced Subgraph: ___________________________________________
________________________________________________________________
________________________________________________________________
3.
Vertex connectivity: ___________________________________________
________________________________________________________________
________________________________________________________________
4.
Chromatic number: ___________________________________________
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________________________________________________________________
5.
Planar graph: ________________________________________________
________________________________________________________________
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1
CSE 8355
Graph Algorithms & Applications
Sample Quiz
D. W. Matula
II. [3 points each]: For each of the terms below, give the notation that
corresponds to the term as shown in the example.
Notation
0.
(Example) A cycle of length n.
___Cn_____
1.
The number of edges in the graph G=(V,E).
__________
2.
A path of length n.
__________
3.
A complete bipartite graph.
__________
4.
The complement of graph G.
__________
5.
The maximum over all subgraphs of the minimum degree of __________
each subgraph.
6.
The graph obtained by deleting edge e from G.
__________
7.
The edge connectivity of graph G.
__________
2
CSE 8355
Graph Algorithms & Applications
Sample Quiz
D. W. Matula
III. [6 points each] There are a number of important theorems in graph theory that
you should be able to recall when needed for applications. Each theorem exhibits
an important relation between certain properties of graphs. Some of these should
be known by the relations they give between properties, and some are simply
identified with important contributors to the field. Indicate any exceptions in the
statement of the theorem.
1.
Relation of maximum clique size to the chromatic number:
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
2.
Euler Polyhedron Formula:
________________________________________________________
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________________________________________________________
3.
Kuratowski’s Theorem:
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
4.
Max flow – min cut Theorem:
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
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3
CSE 8355
Graph Algorithms & Applications
Sample Quiz
D. W. Matula
IV.[5 points each] Give the best (big Oh) algorithm time complexity bound that
you are aware of for each of the following graph problems and give a couple of
key words identifying a method to achieve the time complexity behavior. The time
complexity bound should be for the worst case.
1.
Determining if a graph can be 2-colored:
________________________________________________________
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2.
Determining the diameter of a tree:
________________________________________________________
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3.
Finding an Eulerian tour of a graph:
________________________________________________________
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________________________________________________________
________________________________________________________
4.
Determining the number of faces in an embedding of a planar graph G
having n vertices and m edges:
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
4
CSE 8355
Graph Algorithms & Applications
Sample Quiz
D. W. Matula
V.[5 points each]. For each of the following construct a graph satisfying the
specified properties or give a reason why no such graph exists.
1. A smallest non-trivial tree having no isomorphism with itself other than the
identity.
2, A 3-regular graph with no subgraph of edge connectivity 3.
3. A graph with no complete subgraph of size 3 that has chromatic number 3.
5