Flame holes and flame disks in a laminar axisymmetric counterflow configuration are numerically investigated for unity Lewis number, with the strain rate as the control parameter. The temporal evolution of the topological structure of flame holes and flame disks is described in detail for different representative strain rates. It is found that corresponding to each given strain rate, there exists a critical hole (disk) radius $r_c$ that separates the shrinking and expanding hole (disk) regimes. The value of $r_c$ decreases monotonically with the increase (decrease) of strain rate and reaches a finite minimum at the extinction (ignition) limit of the strain rate, which indicates that one cannot ignite a mixing layer by an infinitesimal energy source, nor can one quench a diffusion flame by making an infinitesimal extinction hole on it. An examination of the phase diagrams of flame holes (disks) justifies the existence of a unique edge-flame velocity $v_f$ as a smooth continuous function of the hole (disk) radius $r_f$ in the entire range $0 < r_f < \infty$, with the strain rate (or equivalently, Damköhler number) as a parameter. For the flame hole case, it is found that in the final stage of collapse of a hole, the edge-flame velocity is essentially proportional to the inverse of the hole radius, except when the strain rate is very close to the extinction limit. Flame interactions induced by overlapping of pre-heat zones are mainly responsible for the acceleration of the edge-flame velocity when the hole radius approaches zero, and it is further enhanced by the focusing effects of hole curvature in the plane of the stoichiometric surface. For the flame disk, the increasing heat loss rate plays a major role on the acceleration of the shrinking speed when the disk radius approaches zero.