1. **Problem Statement:** Identify which angle among \(\angle DOB\), \(\angle BAC\), \(\angle EAC\), and \(\angle ABE\) is a central angle in the given circle. 2. **Definition:** A central angle is an angle whose vertex is at the center of the circle and whose sides (rays) intersect the circle. 3. **Given:** The circle is centered at point \(O\). Points \(D\) and \(B\) lie on the circle such that \(DO\) and \(OB\) form a diameter. Points \(A\) and \(E\) lie on the circumference. 4. **Analysis:** - \(\angle DOB\) has vertex at \(O\), the center of the circle, and its sides \(OD\) and \(OB\) intersect the circle. - \(\angle BAC\), \(\angle EAC\), and \(\angle ABE\) have vertices on the circumference or outside the center. 5. **Conclusion:** By definition, the central angle must have its vertex at the center \(O\). Therefore, \(\angle DOB\) is the central angle. **Final answer:** \(\boxed{\angle DOB}\)