Question

The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex. A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex. In Exercises 39-41 find every vertex that is a center in the given tree.

Short answer

Here \(c\) is the only vertex in the tree having a minimal eccentricity. Hence, \(c\) is the center of the tree.

Step-by-step solution

Definition

The eccentricity of a vertex is the length of the longest simple path starting at the vertex in the unrooted tree.

A vertex is called a center if no vertex within the tree has a smaller eccentricity than this vertex.

Eccentricity of a vertex

In the given tree,

Eccentricity of \(a = 4\)

Eccentricity of \(b= 5\)

Eccentricity of \(c = 3\)

Eccentricity of \(d = 4\)

Eccentricity of \(e = 4\)

Eccentricity of \(f = 5\)

Eccentricity of \(g = 5\)

Eccentricity of \(h = 5\)

Eccentricity of \(i = 5\)

Eccentricity of \(j = 6\)

Eccentricity of \(k = 6\)

Eccentricity of \(l = 6\)