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 g a 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 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. v=center w=centroid