intro to graph theory

8/13/2019 Intro to Graph Theory

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Graph theory? Has nothing to do with graph or graphics

An area of math dealing with entities (nodes) andthe connections (links) between the nodes


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A graph is an abstract mathematicalstructure defined from two sets:

V={n1, n2,nm} of nodes E={e1,e2,em} of edges


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n1

n6

n7

n3

n5n4

n2e1

e5

e3

e4e2


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The degree Refers to the number of edges that have a node

as an endpoint , denoted by deg(n)

Indicates the extent of integration testing that isappropriate for the object

E.g. deg(n1) =2


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Alternative to visual presentation ofgraph

The incidence matrix of G=(V,E) with mnodes and n edges is an mn matrix

We have 1 in row i, column J if node i is an endpoint of edge j

Row sum represents degree of nodes Column sum represents the endpointsof an

edge


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e1 e2 e3 e4 e5

n1 1 1 0 0 0

n2 1 0 0 1 0

n3 0 0 1 0 0

n4 0 1 1 0 1

n5 0 0 0 1 0n6 0 0 0 0 1

n7 0 0 0 0 0


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A useful supplement to the incidencematrix

The Adjacency matrix of G=(V,E) with m

nodes and n edges is an mm matrix We have 1 in row i, and col. j if

there is an edge between node i and node j,

zero otherwise

Used to identify paths and henceequivalence relation to simplify a graphand hence testing


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n1 n2 n3 n4 n5 n6 n7

n1 0 1 0 1 0 0 0

n2 1 0 0 0 1 0 0

n3 0 0 0 1 0 0 0

n4 1 0 1 0 0 1 0

n5 0 1 0 0 0 0 0n6 0 0 0 1 0 0 1

n7 0 0 0 0 0 0 0


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A path is a sequenceof edges such that,for any adjacent pairof edges ei, ej in the

sequence, the edgesshare a common(node) endpoint

Can be described assequences of edgesor nodes

path Nodes

sequence

Edge

sequences

Between

n1 and n5

n1,n2,n5 e1,e4

Between

n6 and n5

n6, n4, n1,

n2, n5

e5,e2,e1,e4

.


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Nodes niand njare connected if they arein the same path

Connectedness is an equivalence

relation can be checked easily Reflexive (every node is in path of 0 length with

itself)

Symmetric n1, and n2 in same path, then n2 and

n1 is also in the same path transitive


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Connectedness defines a partition (orcomponent) on the node set of a graph Components of a graph is maximal set of

connected nodes E.g. Components

S1={n1,n2,n3,n4,n5,n6} and S2={n7}

Condensation graph Used as a Simplification mechanism Creating a graph by replacing a set of

connected nodes (or components) by acondensing node

The implication for testing is that component

are stand alone elements and hence can betested separately


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No edges can be present in a condensationgraph of an ordinary graph.

Two reasons: Edges have individual nodes as endpoints, not sets

of nodes

A possible edge would mean that nodes from twodifferent components are connected, thus in a

path, thus in the same component.

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A directed graph D = (V,E) consists of afinite set V = {n1,., nm} of nodes, and a setE = {e1, e2, ,ep}, where each edge ek =

is an ordered pair of nodes.

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n1

n7

n3

n5n4

n2

n6

e1

e4

e5

e3

e2


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The in degree of a node in a directed graphis the number of distinct edges that havethe node as a terminal node.

The out-degree of a node in a directedgraph is the number of distinct edges that

have the node as a start point.

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The AMD of G=(V,E) with m nodes is anmm matrix where a(i,j) is a 1 if there is

an edge from node i to node j, otherwiseit is 0 Row sum represents outdegrees

Column sum represents indegrees


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n1 n2 n3 n4 n5 n6 n7

n1 0 1 0 1 0 0 0

n2 0 0 0 0 1 0 0n3 0 0 0 1 0 0 0

n4 0 0 0 0 0 1 0

n5 0 0 0 0 0 0 0n6 0 0 0 0 0 0 0

n7 0 0 0 0 0 0 0


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Source node: a node with in-degreezero

Sink node: a node with out-degree=0

Transfer node: node with in-degree 0 and out-degree 0


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Direction is important therefore Directed path (a sequence of edges eiand ej, the

terminal node of eiis the initial node of ej)

Cycle (directed path that begins and ends at thesame node)

Directed semi-path (for adjacent pair of ei, the initial(terminal) node of the first edge is the initial(terminal) node of the second edge

E.g., n1 and n3


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n1 n2 n3 n4 n5 n6 n7

n1 1 1 0 1 1 1 0

n2 0 1 0 0 1 0 0n3 0 0 1 1 0 1 0

n4 0 0 0 1 0 1 0

n5 0 0 0 0 1 0 0n6 0 0 0 0 0 1 0

n7 0 0 0 0 0 0 1


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Connectedness of directed graph 0-connected (no path between ni, and nj)

1-connected (semi-path between ni

, and nj

)

2-connected(a path between ni, and nj )

3-connected (a path between nito nj, and apathbetween nj, and ni)

Strong components


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n1

n7

n3

n5n4

n2

n6

e1

e4

e5

e3

e2

e6

n1 and n7 0-c

n2 and n4 1-c

n1 and n6 2-c

n3 and n6 3-c

intro to graph theory

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