Ch7. Graphs and Trees
ECE20042
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CH. 7. Graphs and Trees (p. 505)
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7.1 Graph Theory(p. 506)
Origins and Euler:
The branch of geometry that deals with
magnitudes has been zealously studied
throughout the past, but there is another
branch that has been almost unknown up until
now; Leibniz spoke of it first, calling it the
“geometry of position.” (geometria situs).
This branch of geometry deals with the
relations dependent on position alone; it does
not take magnitudes into consideration, nor
does it involve calculation of quantities.
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Space must exist independently of object ?
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Space must not exist independently of object ?
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Space must exist independently of object ?
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Graph Theory (p. 506)
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(p. 506)
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Proposition 1 (p. 507)
In any graph, if there are an odd number of
edges connected to a vertex x, then x cannot be
an interior vertex( i.e. a vertex other than the
starting or stopping point) in an Eulerian trail.
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Definition (p. 508)
• A graph G consists of two sets V and E. The
elements of V are called vertices(or nodes),
and the elements of E are called edges. Each
edge is associated with one or two vertices,
called its endpoints. In the diagram we draw
the edge as a line segment or curved are
joining the endpoints.
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Definition (p. 508)
• If two edges have the same endpoints, they are
called multiple edges or parallel edges.
• Two nodes that are joined by an edge are said
to be adjacent nodes.
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Definition (p. 508)
• A walk in a graph is a sequence
v1e1v2e2…vnenvn+1 with n≥ 0 of alternating
vertices and edges, which begins and ends
with a vertex and where each edge in the list
lies between its endpoints. If the beginning
vertex is the same as the ending vertex, we say
the walk is closed. The length of a walk is the
number of edges in the walk. A walk of length
0 is called a trivial walk.
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Definition (p. 508)
• A trail is a walk with no repeated edges, and a
path is a walk with no repeated vertices. A
circuit is a closed trail, and a trivial circuit is
a circuit with one vertex and no edges. A trail
or circuit is called Eulerian if it uses every
edge in the graph.
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Definition (p. 508)
• A cycle is a nontrivial circuit in which the only
repeated node is the first/last one.
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Graphs in Applications (p. 509)
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Definition (p. 510)
• A simple graph is a graph with no loops and no multiple
(parallel) edges.
• The unordered list notation [a,b] indicate an edge with
endpoints a and b. This notation is ambiguous if the graph has
multiple edges, but we will still use it if doing so will not cause
confusion. In a simple graph, an edge with endpoints a and b
can be represented as the two-element set {a,b}
• If the graph is directed, we use (a,b) rather than [a,b] to
indicate a directed edge from a to b. This use of the usual
ordered pair notation emphasizes that in a directed graph the
order of the vertices connected by the edges is significant.
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(p. 510)
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Definition (p. 510)
• An edge e is said to be incident with a node v
if v is an endpoint of e.
• The degree of a node v, denoted by deg(v), is
the number of times v appears as an endpoint
of an edge. That is, deg(v) is the number of
edges that are incident with v, except that
loops are counted twice.
• A graph G is connected if there is a walk
between any pair of distinct nodes.
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Definition (p. 511)
• A graph H is a subgraph of a graph G if all
nodes and edges in H are also nodes and
edges in G.
• A connected component of a graph G is a
connected subgraph H of G such that no other
connected subgraph of G containing H exists.
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Eulerian Graphs (p. 511)
• Definition: A graph G is Eulerian if there is a
circuit in G that involves every edge exactly
once. Recall that such a circuit is called an
Eulerian circuit.
• Theorem 2: Let G be a connected graph. The
Graph G is Eulerian if and only if every node
in G has even degree.
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Example 4, (p. 512)
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Example 4 (p. 512)
• Let G be the graph given on the left in figure
7-11.
– First we find any circuit C, as shown in Figure 711 on the right.
– We next form the graph G' by removing from G
all the edges in the circuit C.
– We can now piece together C, C1, and C2.
Specially, in C=1,2,3,7,8,11,14,1, we replace the 3
with C1 and the 8 with C2 to get the Eulerian
circuit:
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Example 4 (p. 512)
• 1,2,3,4,5,6,7,11,12,13,6,3,11,13,14,3,7,8,9,10,8,11,14,1
C1
C2
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Graphs with Eulerian Trails (p. 513)
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Graphs with Eulerian Trails (p. 513)
• Theorem 3 : In any graph, the sum of the
degrees of the vertices is equal to twice the
number of edges. In symbols.
n
 degv   2m
i 1
i
where v1, v2,…vn are the vertices of the graph
and m is the number of edges in the graph.
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(p. 513)
• Corollary 4: In any graph G, the number of
nodes with odd degree is even.
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(p. 514)
• Theorem 5 : A connected, non-Eulerian graph
G has an Eulerian trail if and only if G has
exactly two nodes of odd degree. Moreover,
the trail must begin and end at these two
nodes. (ref. Theorem 2, p. 511)
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Example 5 (p. 514)
• The graph of the “envelop” we saw earlier can
thus be modified by adding an edge between its
two nodes (3 and 4) of odd degree to get the
graph shown in Figure 7-15. The solution
method of Example 4 can then be used to find
the Eulerian circuit 3,2,1,5,4,6,5,2,6,3,4,3 in
this new graph. Dropping the last edge [4,3]
from this circuit gives us the Eulerian trail
3,2,1,5,4,6,5,2,6,3, 4 in the original graph.
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3,2,1,5,4,6,5,2,6,3,4,3
3
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Example 6 (p. 514)
•
Here is a magic trick based on Eulerian graphs. The explanation is left as an
exercise for the reader. You are the magician. Take a standard set of dominoes and
discard the double 6, double 5, double 4, double 3, double 2, double 1, and double
blank pieces. Have one spectator choose a single domino and give it to you. Turn
your back and have a second spectator line up the remaining dominoes using the
conventional rules that the faces touching must share the same number. The
spectator should only have to make one such line-allow other spectators to help if
he or she is having difficulty. Now you reveal (dramatically) the numbers on the
ends of the line, and show that they are the same two numbers on the domino
originally chosen.
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Eulerian Path & Circuit
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