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 39-41find every vertex that is a center in the given tree.

39.

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 \(b=5\)

Eccentricity of \(b=5\)

Eccentricity of \(i=5\)

Eccentricity of \(j=6\)

Eccentricity of \(k=6\)

Eccentricity of \(l=6\)