Graphs

CSCI 2720Spring 2005

Graph

Why study graphs? important for many real-world

applications compilers Communication networks Reaction networks & more

The Graph ADT a set of nodes (vertices or points) connection relations (edges or arcs)

between those nodes

Definitions follow ….

Definition : graph A graph G=(V,E) is

a finite nonempty set V of objects called vertices (the singular is vertex)

together with a (possibly empty) set E of unordered pairs of distinct vertices of G called edges.

Some authors call a graph by the longer term ``undirected graph'' and simply use the following definition of a directed graph as a graph. However when using Definition 1 of a graph, it is standard practice to abbreviate the phrase ``directed graph'' (as done below in Definition 2) with the word digraph.

Definition: digraph A digraph G=(V,E) is

a finite nonempty set V of vertices together with a (possibly empty) set E of

ordered pairs of vertices of G called arcs.

An arc that begins and ends at a same vertex u is called a loop. We usually (but not always) disallow loops in our digraphs.

By being defined as a set, E does not contain duplicate (or multiple) edges/arcs between the same two vertices.

For a given graph (or digraph) G we also denote the set of vertices by V(G) and the set of edges (or arcs) by E(G) to lessen any ambiguity.

Definition: order, size The order of a graph (digraph)

G=(V,E) is |V|, sometimes denoted by |G| , and the size of this graph is |E| .

Sometimes we view a graph as a digraph where every unordered edge (u,v) is replaced by two directed arcs (u,v) and (v,u) . In this case, the size of a graph is half the size of the corresponding digraph.

Example G1 is a graph of order 5 G2 is a digraph of order

5 The size of G1 is 6

where E(G1) = {(0, 1), (0, 2), (1, 2), (2,

3), (2, 4), (3, 4)} The size of the digraph

G2 is 7 where E(G2) = {(0, 2), (1, 0), (1, 2), (1,

3), (3, 1), (3, 4), (4, 2)}.

Definition: walk, length, path, cycle A walk in a graph (digraph) G is

a sequence of vertices v0, v1, … vn such that, for all 0 <= i< n , (vi, vi+1) is an edge (arc) in G .

The length of the walk v0, v1, … vn is the number n (i.e., number of edges/arcs).

A path is a walk in which no vertex is repeated.

A cycle is a walk (of length at least three for graphs) in which v0

=vn and no other vertex is repeated; sometimes, if it is understood, we omit vn from the sequence.

Example walks Walks in G1:

0,1,2, 3, 4 0,1,2,0 0,1,2 0,1,0

Walks in G2: 3,1,2 1,3,1 3,1,3,1,0

Example paths Paths in G1:

0,1,2, 3, 4 0,1,2

Paths in G2: 3,1,2

Example cycles Cycles in G1:

0,1,2,0 0,1,2 (understood)

Cycles in G2: 1,3,1

Definition: connected, strongly connected

A graph G is connected if there is a path between all pairs of

vertices u and v of V(G) . A digraph G is strongly connected

if there is a path from vertex u to

vertex v for all pairs u and v in V(G).

Connected? G1 is connected

G2 is not strongly connected. No arcs leaving

vertex 2

Definition: degree In a graph, the degree of a vertex v ,

denoted by deg(v), is the number of edges incident to v . in-degree == out-degree

For digraphs, the out-degree of a vertex v is the number of arcs {(v,z) € E| z € V}

incident from v (leaving v ) and the in-degree of vertex v is the number of arcs {(z,v) € E| z € V} incident to v (entering v ).

Degree, degree sequence G1:

deg(0) = 2 deg(1) = 2 deg(2) = 4 deg(3) = 2 Deg(4) = 2

Degree sequence = (2,2,4,2,2)

Degree, degree sequence G2:

In-degree sequence = (1,1,3,1,1)

Out-degree sequence = (1,3,0,2,1)

Degree of vertex of a digraph sometimes written as sum of in-degree and out-degree:

(2,4,3,3,2)

Definition: diameter

The diameter of a connected graph or strongly connected digraph G=(V,E) is the least integer D such that for all

vertices u and v in G we have d(u,v) <=D, where d(u,v) denotes the distance from u to v in G, that is, the length of a shortest path between u and v.

Diameter G1:

min(d(u,v) )= 2 Diameter = 2

G2: not strongly connected, diameter not defined

Computer representations adjacency matrices

For a graph G of order n , an adjacency matrix representation is a boolean matrix (often encoded with 0's and 1's) of dimension n such that entry (i,j) is true if and only if edge/arc (I,j) is in E(G).

adjacency lists For a graph G of order n , an adjacency lists

representation is n lists such that the i-th list contains a sequence (often sorted) of out-neighbours of vertex i of G .

Adjacency matrices for G1,G2

Adjacency lists for G1,G2

For digraphs, stores only the out-edges

Matrix vs. list representation Matrix

n vertices and m edges requires O( n2 ) storage check if edge/arc (i,j) is in graph – O(1)

List n vertices and m edges, requires O(m) storage Preferable for sparse graphs tcheck if edge/arc (i,j) is in graph - O(n) time

Note: other specialized representations exist