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Transcript of CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.
![Page 1: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/1.jpg)
CSE 211Discrete Mathematics
Chapter 8.4Connectivity
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Paths in Undirected Graphs
• There is a path from vertex v0 to vertex vn if there is a sequence of edges from v0 to vn
– This path is labeled as v0,v1,v2,…,vn and has a length of n.
• The path is a circuit if the path begins and ends with the same vertex.
• A path is simple if it does not contain the same edge more than once.
![Page 3: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/3.jpg)
Graph Theoretic Foundations
• Paths and Cycles:
• Walk in a graph G is v0, e1, v1, …, vl-1, el, vl
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Graph Theoretic Foundations
• Paths and Cycles:
• IF v0, v1, …, vl are distinct (except possible v0,vl) then the walk is a path.
• Denoted by v0, v1, …, vl or e1, e2, …, el
• Length of path is l
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Paths, Cycles, and Trails
• A trail is a walk with no repeated edge.
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Graph Theoretic Foundations
• Paths and Cycles:
• A path or walk is closed if v0 = vl
• A closed path containing at least one edge is called a cycle.
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Paths in Undirected Graphs
• A path or circuit is said to pass through the vertices v0, v1, v2, …, vn or traverse the
edges e1, e2, …, en.
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Example
• u1, u4, u2, u3
– Is it simple? – yes
– What is the length? – 3
– Does it have any circuits? – no
u1 u2
u5 u4 u3
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Example
• u1, u5, u4, u1, u2, u3
– Is it simple?– yes
– What is the length?– 5
– Does it have any circuits?– Yes; u1, u5, u4, u1
u1 u2
u5 u3
u4
![Page 10: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/10.jpg)
Example
• u1, u2, u5, u4, u3
– Is it simple?– yes
– What is the length?– 4
– Does it have any circuits?– no
u1 u2
u5 u3
u4
![Page 11: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/11.jpg)
Connectedness
• An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph.
• There is a simple path between every pair of distinct vertices of a connected undirected graph.
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Example
Are the following graphs connected?
c
e
f
a
d
b
g
e
c
a
f
b
d
Yes No
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Connectedness (Cont.)
• A graph that is not connected is the union of two or more disjoint connected subgraphs (called the connected components of the graph).
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Example
• What are the connected components of the following graph?
b
a c
d e
h g
f
![Page 15: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/15.jpg)
Example
• What are the connected components of the following graph?
b
a c
d e
h g
f
{a, b, c}, {d, e}, {f, g, h}
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Cut edges and vertices
• If one can remove a vertex (and all incident edges) and produce a graph with more connected components, the vertex is called a cut vertex.
• If removal of an edge creates more connected components the edge is called a cut edge or bridge.
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Example
• Find the cut vertices and cut edges in the following graph.
a
b c
d f
e h
g
![Page 18: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/18.jpg)
Example
• Find the cut vertices and cut edges in the following graph.
a
b c
d f
e h
g
Cut vertices: c and eCut edge: (c, e)
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Graph Theoretic Foundations• Connectivity:• Connectivity (G) of graph G is …• G is k connected if (G) k• Separator or vertex-cut Cut
vertex
Separation pair
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Graph Theoretic Foundations
• Trees and Forests:• Tree – connected graph without any cycle• Forest – a graph without any cycle
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Connectedness in Directed Graphs
• A directed graph is strongly connected if there is a directed path between every pair of vertices.
• A directed graph is weakly connected if there is a path between every pair of vertices in the underlying undirected graph.
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Example
• Is the following graph strongly connected? Is it weakly connected?
a b
c
e d
This graph is strongly connected. Why? Because there is a directed path between every pair of vertices.
If a directed graph is strongly connected, then it must also be weakly connected.
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Example
• Is the following graph strongly connected? Is it weakly connected?
a b
c
e d
This graph is not strongly connected. Why not? Because there is no directed path between a and b, a and e, etc.
However, it is weakly connected. (Imagine this graph as an undirected graph.)
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Connectedness in Directed Graphs
• The subgraphs of a directed graph G that are strongly connected but not contained in larger strongly connected subgraphs (the maximal strongly connected subgraphs) are called the strongly connected components or strong components of G.
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Example
• What are the strongly connected components of the following graph?a b
c
e d
This graph has three strongly connected components:• The vertex a• The vertex e• The graph consisting of V = {b, c, d} and E = { (b, c), (c, d), (d, b)}
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Consider a connected graph G. The distance between vertices u and v in G, written d(u, v), is the length of the shortest path between u and v. The diameter of G, written diam(G), is the maximum distance between any two points in G. For example, in Fig. 1-8(a), d(A,F) = 2 and diam(G) = 3, whereas in Fig. 1-8(b), d(A, F) = 3 and diam(G) = 4
Distance in Trees and Graphs
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Distance in Trees and Graphs
• If G has a u, v-path, then the distance from u to v, written dG(u, v) or simply d(u, v), is the least length of u, v-path.
• The diameter (diam G) is maxu,vV(G)d(u,v)
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Distance in Trees and Graphs
• Find the diameter, eccentricity, radius and center of the given G.
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CSE 211 Discrete Mathematics and Its Applications
Chapter 8.5Euler and Hamilton Paths
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Euler Paths and Circuits
• The Seven bridges of Königsberg
a
b
c
dA
B
C
D
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Euler Paths and Circuits
• An Euler path is a path using every edge of the graph G exactly once.
• An Euler circuit is an Euler path that returns to its start.
A
B
C
DDoes this graph have an
Euler circuit?No.
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Necessary and Sufficient Conditions
• How about multigraphs?
• A connected multigraph has a Euler circuit iff each of its vertices has an even degree.
• A connected multigraph has a Euler path but not an Euler circuit iff it has exactly two vertices of odd degree.
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Example• Which of the following graphs has an Euler circuit?
e
d
a
c
b
e
d
a
c
b
ec
a
d
b
yes no no(a, e, c, d, e, b, a)
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Example• Which of the following graphs has an Euler path?
e
d
a
c
b
e
d
a
c
b
ec
a
d
b
yes no yes(a, e, c, d, e, b, a ) (a, c, d, e, b, d, a, b)
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Euler Circuit in Directed Graphs
NO (a, g, c, b, g, e, d, f, a) NO
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Euler Path in Directed Graphs
NO (a, g, c, b, g, e, d, f, a) (c, a, b, c, d, b)
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Hamilton Paths and Circuits
• A Hamilton path in a graph G is a path which visits every vertex in G exactly once.
• A Hamilton circuit is a Hamilton path that returns to its start.
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Hamilton Circuits
Is there a circuit in this graph that passes through each vertex exactly once?
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Hamilton Circuits
Yes; this is a circuit that passes through each vertex exactly once.
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Finding Hamilton Circuits
Which of these three figures has a Hamilton circuit? Of, if no Hamilton circuit, a Hamilton path?
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Finding Hamilton Circuits
• G1 has a Hamilton circuit: a, b, c, d, e, a• G2 does not have a Hamilton circuit, but does have a Hamilton path: a, b, c, d• G3 has neither.
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Finding Hamilton Circuits
• Unlike the Euler circuit problem, finding Hamilton circuits is hard.
• There is no simple set of necessary and sufficient conditions, and no simple algorithm.
![Page 43: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/43.jpg)
Properties to look for ...
• No vertex of degree 1• If a node has degree 2, then both edges
incident to it must be in any Hamilton circuit.
• No smaller circuits contained in any Hamilton circuit (the start/endpoint of any smaller circuit would have to be visited twice).
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A Sufficient Condition
Let G be a connected simple graph with n vertices with n 3.
G has a Hamilton circuit if the degree of each vertex is n/2.
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Travelling Salesman Problem
A Hamilton circuit or path may be used to solve practical problems that require visiting “vertices”, such as:
road intersectionspipeline crossingscommunication network nodes
A classic example is the Travelling Salesman Problem – finding a Hamilton circuit in a complete graph such that the total weight of its edges is minimal.
![Page 46: CSE 211 Discrete Mathematics Chapter 8.4 Connectivity.](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515190055034673228b4d5f/html5/thumbnails/46.jpg)
Summary
Property Euler Hamilton
Repeated visits to a given node allowed?
Yes No
Repeated traversals of a given edge allowed?
No No
Omitted nodes allowed? No No
Omitted edges allowed? No Yes