Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A...
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Transcript of Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A...
![Page 1: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/1.jpg)
Graph Theory
Introducton
![Page 2: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/2.jpg)
Graph Theory
Vertex: A point. An intersection of two lines (edges).
Edge: A line (or curve) connecting two vertices.
Loop: An edge that connects a vertex to itself only.
![Page 3: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/3.jpg)
Graph TheoryEx) Represent the "Konigsberg Bridge“
problem using a vertex-edge graph.
* Vertices represent locations.* Edges represent “connections” between those locations.
A
B C
D
A
B C
D
![Page 4: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/4.jpg)
Graph TheoryEx) Represent this map using a vertex-edge graph.
Hint: On map problems, place vertices relative to their actual locations on the map.
O
K
CU
W
N* Edges represent borders in a map problem.
![Page 5: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/5.jpg)
Graph TheoryEx) Represent a floor plan using a vertex-edge graph.
Outside P
HF
J
C
M
O
![Page 6: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/6.jpg)
Graph TheoryThe degree of a vertex is the number of edges "entering" the vertex.
Degree2 1
2
Degree3
12 3 Degree
41
2 3
4
![Page 7: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/7.jpg)
Graph TheoryOdd and Even vertices
If the degree of a vertex is an odd number, then the vertex is considered an odd vertex.
If the degree of the vertex is an even number, then the vertex is considered an even vertex.
![Page 8: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/8.jpg)
Graph TheoryEx) How many odd vertices are there in this graph?
Degree4
Degree4
Degree3
Degree4
Degree1
2 odd vertices F
Degree0
![Page 9: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/9.jpg)
Graph TheoryA path is a sequence of adjacent vertices and the edges connecting them.
Given the graph to the left, some examples of paths could be:
ABC
BCDD
CDBCA
![Page 10: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/10.jpg)
Graph TheoryA circuit is a path that begins and ends at the same vertex.
Given the graph to the left, some examples of circuits could be:
ABCACDDBAC
![Page 11: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/11.jpg)
Graph TheoryOn a connected graph, you can draw a path from one vertex to any other vertex.
![Page 12: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/12.jpg)
Graph TheoryIf a graph is not connected, it is disconnected.
![Page 13: Graph Theory Introducton Graph Theory Vertex:A point. An intersection of two lines (edges). Edge:A line (or curve) connecting two vertices. Loop:An edge.](https://reader035.fdocuments.net/reader035/viewer/2022081504/5697bf7a1a28abf838c82f35/html5/thumbnails/13.jpg)
Graph TheoryA bridge is an edge that if removed from a connected graph would create a disconnected graph.
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Graph Theory
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