Star Graph - The Trick

Look at Edge $1$: [u1, v1]. Look at Edge $2$: [u2, v2]. One of the numbers in Edge $1$ must be the center. One of the numbers in Edge $2$ must also be the center.

The center is the common node. Compare u1 with u2 and v2. If u1 equals either, return u1. Otherwise, return v1.

if u1 = u2 or u1 = v2 then
    return u1
else
    return v1

This runs in constant time regardless of graph size. Two edges are enough to identify the unique center.