MATHEMATICS 2P GRAPHS AND NETWORKS
EXERCISE SHEET 2
1. Prove that the utilities graph is not planar by imitating the proof for the complete graph on ve vertices.
2. Show that if G is a simple connected planar bipartite graph with p 3 then q 2p 4. (It may
MATHEMATICS 2P GRAPHS AND NETWORKS
EXERCISE SHEET 1
1. Draw
(a) all six non-isomorphic connected simple graphs with 4 vertices;
(b) all non-isomorphic connected simple graphs with p = q = 5;
(c) all non-isomorphic trees with 6 vertices.
2. Explain why eve
Mathematics 2P Graphs and networks
10
3. BIPARTITE GRAPHS
Denition
A graph is bipartite if its vertices can be partitioned into two sets V1 and V2 such that every edge joins
a vertex in V1 to a vertex in V2 . (Every vertex must be in V1 or V2 , and no ver
MATHEMATICS 2P GRAPHS AND NETWORKS
SOLUTIONS TO EXERCISE SHEET 2
1. The utilities graph is the complete bipartite graph K3,3 , and it is a non-empty connected simple
graph. Suppose that it is planar with p vertices and q edges, and suppose that a plane dr
Mathematics 2P Graphs and networks
28
8. FLOWS IN NETWORKS
Introduction
We consider directed graphs in which each edge is assigned a nonnegative number, called its capacity.
To object is to get as much as possible of some commodity to ow through the netwo
Mathematics 2P Graphs and networks
5
2. TREES
Denition
Recall that a tree is a non-empty connected graph containing no cycles.
Paths in trees
Theorem 2.1. In a tree there is a unique path between any two vertices.
Proof. Since a tree is connected, there i
Mathematics 2P Graphs and networks
42
11. CRITICAL PATH SCHEDULING
Introduction
We continue to study projects consisting of activities with various durations and precedence requirements. We want to complete the project as quickly as possible using a limit
Mathematics 2P Graphs and networks
23
6. EULERIAN GRAPHS
The Knigsberg bridges problem
o
In Knigsberg in 1738 there were islands and bridges in the river Pregel as shown.
o
A
B
C
D
The inhabitants wished to go for walks starting and nishing at the same po
Mathematics 2P Graphs and networks
18
5. HAMILTONIAN GRAPHS
Introduction
Recall that a hamiltonian graph is a graph with a hamiltonian cycle, i.e. a cycle passing through all the
vertices.
For example the rst graph shown below is hamiltonian, with hamilto
1
MATHEMATICS 2P GRAPHS AND NETWORKS
SOLUTIONS TO EXERCISE SHEET 1
1. (a)
(b)
(c)
2. The sum of the degrees of the vertices is twice the number of edges and is therefore even. It follows
that there must be an even number of vertices with odd degree.
3. (a
1
MATHEMATICS 2P GRAPHS AND NETWORKS
EXERCISE SHEET 3
1. Consider the network ow shown, where labels denote ows and capacities.
4, 6
A
D
7, 13
1, 1
3, 7
3, 7 C 10, 12
S
T
0, 5
12, 15
6, 10
6, 11
6, 9
B
E
(a) Which of the paths
SADCT, SBCET, SBECADT, SACDT
Thursday, 23rd November, 2006 10.10 am. to 10.55 am.
UNIVERSITY OF GLASGOW
Department of Mathematics
Mathematics 2P Graphs and Networks
Class Test
Candidates should attempt ALL questions.
1. Show that the alcohol C4H90H has a tree-like molecule (the valen
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Mathematics 2P Graphs and Networks
Candidates must not attempt more than THREE questions.
1. State the denition of a tree. 2
(ii) Let A and B be vert
1
MATHEMATICS 2P GRAPHS AND NETWORKS
SOLUTIONS TO EXERCISE SHEET 3
1. (a) In SADCT the forward edge DC is saturated, so SADCT is not ow-augmenting. In SBCET
the ows along the forward edges SB, BC, CE and ET can be increased by 3, 4, 5 and 5, all of which
Wednesday, 5th November, 2014 12.10 pm. to 12.55 pm.
University
(f Glasgow
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2P - Graphs and Networks
An electronic calculator may be used provided that it does not have
a facility for either textual
Mathematics 2P Graphs and networks
25
7. DIRECTED GRAPHS
Denition
A directed graph or digraph is a graph in which each edge has a direction. In diagrams, the directions
are indicated by arrows, for example as follows.
b
a
d
e
c
Indegrees and outdegrees
In
Mathematics 2P Graphs and networks
39
10. ACTIVITY NETWORKS
Introduction
In this chapter we apply the longest path algorithm to scheduling problems.
We consider a project consisting of several activities of various durations. There are some precedence
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Mathematics 2P Graphs and networks
35
9. THE SHORTEST AND LONGEST PATH ALGORITHMS
Introduction
In this chapter we mainly consider paths in directed graphs. All these paths are assumed to be directed
paths, i.e. one goes along each edge in the correct dire
Mathematics 2P Graphs and networks
1
MATHEMATICS 2P GRAPHS AND NETWORKS
1. ELEMENTARY RESULTS ON GRAPHS
Denition
A graph is an object consisting of a nite set of vertices and a nite set of edges, such that to each
edge e there is associated a pair of vert
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Mathematics 2P Graphs and networks
11
4. PLANARITY
Denitions
A plane graph is a graph such that: the vertices are distinct points in a plane; the edges are curves
joining the vertices; the edges do not meet the vertices or each other except at their end p