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COSC 2007Data Structures II
Chapter 14Graphs I
2
Topics
Introduction & Terminology ADT Graph
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Introduction Graphs
Important mathematical concept that have significant application in computer science
Can be viewed as a data structure or ADT Provide a way to represent relationships between data
Questions Answered by Using Graphs: Airline flight scheduling:
What is the shortest distance between two cities?
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• A graph G is a pair (V, E) where
V is a set of vertices (nodes) V = {A, B, C, D, E, F}
E is a set of edges (connect vertex) E = {(A,B), (A,D), (B,C),(C,D), (C,E), (D,E)}
A
BC
F
D E
Terminology and Notations
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• If edge pairs are ordered, the graph is directed, otherwise undirected.
• We draw edges in undirected graphs with lines with no arrow heads.
This is an undirected graph.
(B, C) and (C, B) mean the same edge
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BC
F
D E
Terminology and Notations
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• If edge pairs are ordered, the graph is directed, otherwise undirected.
• We draw edges in directed graphs with lines with arrow heads.
A
BC
F
D E
This is adirected graph.
This edge is (B, C).
(C, B) would mean a directed edge from C to B
Terminology and Notations
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• Directed Graph (Digraph):If an edge between two nodes has a direction (directed edges)
Out-degree (OD) of a Node in a Digraph:The number of edges exiting from the node
In-degree (ID) of a Node in a Digraph:The number of edges entering the node
A
BC
F
D E
This is adirected graph.
Terminology and Notations
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• Vertex w is adjacent to v if and only if (v, w) E.
• In a directed graph the order matters:
B is adjacent to A in this graph, but A is not adjacent to B.
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BC
F
D E
Terminology and Notations
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• Vertex w is adjacent to v if and only if (v, w) E.
• In an undirected graph the order does not matter:
we say B is adjacent to A and that A is adjacent to B.
A
BC
F
D E
Terminology and Notations
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• In some cases each edge has a weight (or cost) associated with it.
• The costs might be determined by a cost function
E.g., c(A, B) = 3,c(D,E) = – 2.3, etc.
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A
BC
F
D E
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7.5
– 2.3
1.24.5
Terminology and Notations
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• In some cases each edge has a weight (or cost) associated with it.
• When no edge exists between two vertices, we say the cost is infinite.
E.g., c(C,F) =
A
BC
F
D E
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7.5
– 2.3
1.24.5
Terminology and Notations
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• Let G = (V, E) be a graph.
A subgraph of G is a graph H = (V*, E*) such that V* V and E* E.
A
BC
F
D E
E.g.,
V* = {A, C, D},
E* = {(C, D)}.
Terminology and Notations
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• Let G = (V, E) be a graph.
A subgraph of G is a graph H = (V*, E*) such that V* V and E* E.
AC
D
E.g.,
V* = {A, C, D},
E* = {(C, D)}.
Terminology and Notations
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• Let G = (V, E) be a graph.
A path in the graph is a sequence of vertices
w , w , . . . , w such that (w , w ) E for 1<= i <= N–1. 1 2 N i i+1
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BC
F
D E
E.g., A, B, C, Eis a path inthis graph
Terminology and Notations
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• Let w , w , . . . , w be a path.
The length of the path is the number of edges, N–1, one less than the number of vertices in the path.
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D E
E.g., the length ofpath A, B, C, Eis 3.
1 2 N
Terminology and Notations
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• Let w , w , . . . , w be a path in a directed graph.
Since each edge (w , w ) in the path is ordered,
the arrows on the path are always directed along
the path.
A
BC
F
D E
1 2 N
i i+1
E.g., A, B, C, Eis a path in this directed graph, but . . .
Terminology and Notations
. . . but A, B, C, Dis not a path, since (C, D) is not an edge.
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A path is simple if all vertices in it are distinct, except that the first and last could be the same.
A
BC
F
D E
E.g., thepath A, B, C, Eis simple . . .
Terminology and Notations
. . . and so is thepath A, B, C, E, D, A.
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If G is an undirected graph, we say it is connected if there is a PATH from every vertex to every other vertex.
A
BC
F
D E
This undirected graph is not connected.
Terminology and Notations
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If G is an directed graph, we say it is strongly connected if there is a path from every vertex to every other vertex.
This directed graph is strongly connected.
A
BC
F
D E
Terminology and Notations
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If G is an directed graph, we say it is strongly connected if there is a path from every vertex to every other vertex.
This directed graph is not strongly connected; e.g., there’s no path
from D to A.
A
BC
F
D E
Terminology and Notations
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If G is directed and not strongly connected, but the underlying graph (without direction to the edges) is connected, we say that G is weakly connected.
This directed graph is not strongly connected, but it isweakly connected, since . . .
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D E
Terminology and Notations
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If G is directed and not strongly connected, but the underlying graph (without direction to the edges) is connected, we say that G is weakly connected.
. . . since theunderlying undirected
graph is connected.
A
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D E
Terminology and Notations
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Cycle: a path that begins and ends at the
same node but doesn't pass through other
nodes more than once
A
BC
F
D E
The pathA, B, C, E, D, A
is a cycle.
Terminology and Notations
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•A graph with no cycles is called acyclic.
A
BC
F
D E
This graph is acyclic.
Terminology and Notations
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•A graph with no cycles is called acyclic.
This directed graph is not acyclic, . . .
A
BC
F
D E
Terminology and Notations
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•A graph with no cycles is called acyclic.
. . . but this one is.A Directed Acyclic Graph is often called simply a DAG.
A
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D E
Terminology and Notations
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•A complete graph is one that has an edge between every pair of vertices.
A
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D E
Incomplete:
Terminology and Notations
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•A complete graph is one that has an edge between every pair of vertices. (if the graph contains the maximum possible number of edges) A complete graph is also connected, but the converse is not true
Complete:
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Terminology and Notations
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•A complete graph is one that has an edge between every pair of vertices.
• Suppose G = (V, E) is complete. Can you express |E| as a function of |V|?Complete:
A
BC
D E
This graph has |V| = 5 vertices and |E| = 10 edges.
Terminology and Notations
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•A free tree is a connected, acyclic, undirected graph.
•“Free” refers to the fact that there is no vertex designated as the “root.”
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F
Terminology and Notations
This is a free tree.
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•A free tree is a connected, acyclic, undirected graph.
•If some vertex is designated as the root, we have a rooted tree.
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F
root
Terminology and Notations
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•If an undirected graph is acyclic but possibly disconnected, it is a forest.
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D E
F
GH
I
JL
K
This is a forest. It contains three free trees.
Terminology and Notations
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•If an undirected graph is acyclic but possibly disconnected, it is a forest.This graph contains a cycle. Therefore it is neither a free tree nor a forest.
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GH
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Terminology and Notations
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Review In a graph, a vertex is also known as a(n)
______. node edge path cycle
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Review A graph consists of ______ sets.
two three four five
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Review A subset of a graph’s vertices and edges
is known as a ______. bar graph line graph Subgraph circuit
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Review Two vertices that are joined by an edge
are said to be ______ each other. related to bordering utilizing adjacent to
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Review All ______ begin and end at the same
vertex and do not pass through any other vertices more than once. paths simple paths cycles simple cycles
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Review Which of the following is true about a
simple cycle? it can pass through a vertex more than once it can not pass through a vertex more than
once it begins at one vertex and ends at another it passes through only one vertex
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Review A graph is ______ if each pair of distinct
vertices has a path between them. complete disconnected connected full
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Review A complete graph has a(n) ______
between each pair of distinct vertices. edge path Cycle circuit
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Review The ______ of a weighted graph have
numeric labels. vertices edges paths cycles
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Review The edges in a ______ indicate a
direction. graph multigraph digraph spanning tree
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Review If there is a directed edge from vertex x to
vertex y, which of the following can be concluded about x and y? y is a predecessor of x x is a successor of y x is adjacent to y y is adjacent to x
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