Introduction to Graph Theory

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

Terminology • Eccentricity • If e(v)=t, then the distance from v to any other vertex of G is no more than t • There is at least one vertex whose distance from v is 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.

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 applications? • 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,

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) n b a g c f k t j

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 • …

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

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 path • 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 vice 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 overlapping at uv • Thus diam(T)=2rad(T)-1

Centroid of Tree • Given a vertex v of a tree T, the maximal subtrees that have v as an end vertex are called branchesat 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. v=center w=centroid

Introduction to Graph Theory