COSC 2007 Data Structures II Chapter 14 Graphs I

COSC 2007Data Structures II

Chapter 14Graphs I

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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)}

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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 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


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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.

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BC

F

D E

This is adirected graph.

This edge is (B, C).

(C, B) would mean a directed edge from C to B


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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

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BC

F

D E

This is adirected graph.


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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


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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.

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BC

F

D E


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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

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F

D E

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7.5

– 2.3

1.24.5


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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) =

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BC

F

D E

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7.5

– 2.3

1.24.5


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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.

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D E

E.g.,

V* = {A, C, D},

E* = {(C, D)}.


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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)}.


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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

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D E

E.g., A, B, C, Eis a path inthis graph


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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


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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.

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BC

F

D E

1 2 N

i i+1

E.g., A, B, C, Eis a path in this directed graph, but . . .


. . . 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.

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D E

E.g., thepath A, B, C, Eis simple . . .


. . . 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.

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D E

This undirected graph is not connected.


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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.

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D E


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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.

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F

D E


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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


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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.

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D E


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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

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D E

The pathA, B, C, E, D, A

is a cycle.


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•A graph with no cycles is called acyclic.

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D E

This graph is acyclic.


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This directed graph is not acyclic, . . .

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F

D E


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. . . but this one is.A Directed Acyclic Graph is often called simply a DAG.

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D E


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•A complete graph is one that has an edge between every pair of vertices.

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D E

Incomplete:


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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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D E


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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:

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This graph has |V| = 5 vertices and |E| = 10 edges.


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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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D E

F


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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D E

F

root


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•If an undirected graph is acyclic but possibly disconnected, it is a forest.

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BC

D E

F

GH

I

JL

K

This is a forest. It contains three free trees.


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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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D E

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GH

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JL

K


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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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COSC 2007 Data Structures II Chapter 14 Graphs I