Introduction to Graph Theory

Lecture 08: Distance and Connectivity

Introduction

We have came across the concept of distance when we studied isomorphism.

Many graph algorithms are related to searching for paths of various lengths within the graph.

Connectivity is important since it is strongly related to reliability and vulnerability of computer networks.

Distance

Distance d(u,v) is the number of edges in any u-v geodesic in G.

d(u,v) obeys metric rules: and

, known as triangle inequality

0, vud vuvud 0,

vuuvdvud ,,,

vwdwudvud ,,,

Terminology

Eccentricity If e(v)=t, then the distance from v to any other verte

x of G is no more than t There is at least one vertex whose distance from v i

s t. If d(v,w)=e(v) then w is an eccentric vertex of v

Should v be the eccentric vertex of w? If u and v are eccentric vertices of one another,

they are mutually eccentric.

vudveGVu

,max

Terminology

The minimum eccentricity among the vertices of a graph is called the radius of G, rad(G).

A set of vertices with minimum eccentricity is called the center. What is the importance of the notion of center in application

s? Maximum eccentricity is call the diameter, diam(G).

(What if G is disconnected?) Periphery, P(G), is the set of vertices with maximum

eccentricity. For any graph G, GGG rad2diamrad

Terminology

An antipodal/diametral pair of vertices u and v satisfies d(u,v)=diam(G); each vertex is termed an antipode of the other. Antipodal vertices are always mutually eccentric.

A radial path is a geodesic joining the central vertex to one of its eccentric vertices.

A diametral path is a geodesic joining the diametral pair of vertices.

Exercise (Example 4.1)

g bn

t

j

a

c fk

Theorem on Eccentricity

Theorem: If u and v are adjacent vertices in a connected graph, then . In other words, eccentricities of adjacent vertices differ by at most 1.

Proof: for all x Let w be an eccentric vertex of v, so …

1 veue

uexud , wvdve ,

uewudwuduvdwvdve 1,1,,,

Distance Properties for Trees

P1: Given 2 vertices u, v, and w, such that u and v are adjacent, we have

P2: All eccentric vertices of a tree are end vertices P3: Pairs of antipodal vertices of a tree are end vertic

es P4: The periphery of a tree consists of end vertices P5: In any tree T, every diametral path includes all ce

ntral vertices of T.

1,, vwdvud

The Center of a Tree

Theorem: The center of a tree consists of either a single vertex or two adjacent vertices.

Proof: by pruning We prune the end vertices one layer at a time This decrease the eccentricity of each surviving

vertex by exactly 1 (by P2) Center remains unchanged What remains is either a single vertex or 2 adjacent

vertices.

The Center of a Tree

Theorem: For any tree T, if |C(T)|=1, then diam(T)=2*rad(T), and if |C(T)|=2, then diam(T)=2*rad(T)-1.

Proof: for |C(T)|=1, and let C(T)={v} If V(T)=1, then it is trivial If V(T)>=3, then there are at least 2 branches at v Two such branches must contain radial paths Let x and y be the end vertices of the two radial p

ath The geodesic joining x and y is 2*rad(T)=diam(T)

The Center of a Tree

Proof: for |C(T)|=2, and let C(T)={v,u} From previous theorem, we know that d(u,v)=1 All the radial path starting from u contains v, and v

ice versa. So all the radial paths have uv in common By P5, every diametral x-y path contains uv The x-y path is geodesic x->C(T) + uv + C(T)->y Thus x-y path composed of two radial paths overl

apping at uv Thus diam(T)=2rad(T)-1

Centroid of Tree Given a vertex v of a tree T, the maximal subt

rees that have v as an end vertex are called branches at v.

Weight of a vertex is the largest number of edges among all of its branches.

v v

Number of branches for v is 4, weight is 2

Centroid of Tree

The centroid of a tree T is the set of vertices with minimum weight.

It consisting of a single vertex or two adjacent vertices.

The center and centroid may be disjoint.

w=centroidv=center