Graphs Examples on some basic graph concepts and definitions All graphics are taken from the LEDA demos: basic_graph_algorithms, gw_shortest_paths, graphwin

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GraphsExamples on some basic graph concepts and
definitionsAll graphics are taken from the LEDA
demos basic_graph_algorithms,
gw_shortest_paths, graphwin

• Geetika Tewari
• 252a-al

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Vertex The point of intersection of lines. The
terminating point of some lines in a figure or
curve
Several Vertices
Edge Line that connects 2 vertices
Loop an edge that connects the same two vertices
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Multiple Edges More than one edge connecting two
vertices
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Undirected Graph A graph whose edges are all
undirected
Directed graph A graph whose edges are directed
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Simple Graph A graph that does not contain any
loops or multiple edges
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Examples 1. Example of a graph Notice it is
undirected, has multiple edges and loops
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2. Example of a graph Notice the unconnected
components
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3. Example of a mutigraph (graph with multiple
edges)
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Special Cases of graphs
1. Complete Graph A full graph where every edge
is connected to every other edge
Complete graph of 3 vertices
Complete graph of 11 vertices
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2. Bipartite Graphs A graph whose vertices can
be divided into two classes. Edges exist only
between vertices that belong to different classes.
An example of a complete bipartite graph
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Paths Path in an directed graph a series of
connected vertices
Notice in this example there is a path from
vertex 0 to vertex 4 0, 1, 3, 4, and a path
from vertex 0 to vertex 20, 1, 3, 2
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Path in an undirected graph
In this graph there is a path 11, 5, 7, 13,
10 There are other paths too..
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Hamilton Path in an Undirected Graph A path that
spans all the vertices in a graph
Path 5, 0, 2, 3, 1, 4, 6
Hamilton Path in a Directed Graph A path that
spans all the vertices in the digraph Eg 0, 3,
5, 7, 10, 17, 21, 28
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Cycles In a graph a path Vo, V1, V2Vk forms
a cycle if Vo Vk and if V1, V2.Vk are distinct.
? There is a cycle in this undirected graph 0,
1, 3, 0,
Cycles in this directed graph 2, 3, 4, 2
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Hamilton Cycles A Cycle that covers all the
vertices in a graph
Hamilton Cycle in a directed graph
0, 1, 3, 2, 4, 5, 6, 7, 0
Hamiltion Cycle in an undirected graph 7, 8, 1,
4, 3, 2, 0, 5, 6, 7
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Cyclic and Acyclic Digraphs
A cyclic digraph a directed graph with cycles
0, 3, 6, 0, 0, 1, 2, 4, 5, 3, 6, 0
An acyclic digraph a directed graph with no
cycles
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Tree A connected, acyclic undirected graph
Forest An acyclic undirected graph, i.e. many
trees grouped together
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Articulation Points In an Undirected Graph An
articulation point is a vertex in a graph whose
removal disconnects the graph. In the example
below, the red vertices are the articulation
points.