Graphs KRW

7/28/2019 Graphs KRW

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Graph Algorithms

Kasun Ranga Wijeweera

(Email: [email protected])



What is a Graph?•

Intuitively, a graph is a collection of vertices (or nodes) andthe connections between them

• Generally, no restriction is imposed on the number of vertices

in the graph or on the number of connections one vertex can

have to other vertices



A Simple Graph•

A simple graph G = (V, E) consists of a nonempty set V of vertices and a possibly empty set E of edges, each edge being a

set of two vertices from V

• |V| = Number of vertices

•|E| = Number of edges



A Directed Graph•

A directed graph, or digraph, G = (V, E) consists of anonempty set V of vertices and a set E of edges (also called

arcs), where each edge is pair of vertices from V

• The difference is that one edge of a simple graph is of the

form {vi, v j}, and in this case, (vi, v j) != (v j , vi)



A Multi Graph•

Above definitions are restrictive in that they do not allow for two vertices to have more than one edge

• A multi graph is graph in which two vertices can be joined by

multiple edges

•Formal Definition: A multi graph G = (V, E, f) is composed of a set of vertices V, a set of edges E, and a function

f: E{(vi, v j): (vi, v j in V) and (vi != v j)}



A Pseudo Graph•

A pseudo graph is a multi graph with the condition vi != v j removed, which allows for loops to occur

• In a pseudo graph, a vertex can be joined with itself by an edge



A Path•

A path from v1 to vn is a sequence of edges edge(v1v2),edge(v2v3), . . . , edge(vn-1vn) and is denoted as path v1, v2, v3, .

. . , vn-1, vn

• If v1 = v2 and no edge is repeated, then the path is called a

circuit• If all vertices in a circuit are different, then it is called a cycle



A Weighted Graph•

A graph is called a weighted graph if each edge has anassigned number

• Depending on the context in which such graphs are used, the

number assigned to an edge is called it weight, cost, distance,

length, or some other name



A Sub Graph•

A sub graph G’ of graph G = (V, E) is a graph (V’, E’) suchthat V’ is a subset of V and E’ is a subset of E

• A sub graph induced by vertices V’ is a graph (V’, E’) such

that and edge e in E if e in E’



Adjacent? Incident?•

Two vertices vi and v j are called adjacent if the edge(viv j) is inE

• Such an edge is called incident with the vertices vi and v j



The Degree of a Vertex•

The degree of a vertex v, deg(v), is the number of edgesincident with v

• If deg(v) = 0, then v is called an isolated vertex



Adjacency Matrix•

An adjacency matrix of graph G = (V, E) is a binary |V|*|V|matrix such that each entry of this matrix

a ij

1; if there exists an edge(viv j)

0; otherwise



Graph Traversals•

Traversing a graph consists of visiting each vertex only onetime

• Graphs may include cycles that can cause infinite loops

• To prevent infinite loops each visited vertex can be marked to

avoid revisiting it• Graphs can have isolated vertices

• To visit those isolated vertices special mechanisms are needed

• The Depth First Search algorithm is a well known algorithm

for traversing graphs



Depth First Search Algorithm•

Each vertex v is visited and then each unvisited vertexadjacent to v is visited

• If a vertex v has no adjacent vertices or all of its adjacent

vertices have been visited, we backtrack to the predecessor of

v

• The traversal is finished if this visiting and backtracking

process leads to the first vertex where the traversal started

• If there is still some unvisited vertices in the graph, the

traversal continues restarting for one of the unvisited vertices• The algorithm assigns a unique number to each accessed

vertex so that vertices are now renumbered



Depth First Search Algorithm

depthFirstSearch()for all vertices v

num(v) = 0;

edges = null;

i = 1;

while there is a vertex v such that num(v) is 0

DFS(v);

output edges;



Depth First Search Algorithm

DFS(v)num(v) = i++;

for all vertices u adjacent to v

if num(u) is 0

attach edge(uv) to edges;

DFS(u);



Reference



Any Questions?



Thank You!

Graphs KRW

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