A novel method to characterize the topology of the early-universe intergalactic medium during the epoch of cosmic reionization is presented. The 21-cm radiation background from high redshift is analyzed through calculation of the 2-dimensional (2D) genus. The radiative transfer of hydrogen- ionizing photons and ionization-rate equations are calculated in a suite of numerical simulations under various input parameters. The 2D genus is calculated from the mock 21-cm images of high-redshift universe. We construct the 2D genus curve by varying the threshold differential brightness temperature, and compare this to the 2D genus curve of the underlying density field. We find that (1) the 2D genus curve reflects the evolutionary track of cosmic reionization and (2) the 2D genus curve can discriminate between certain reionization scenarios and thus indirectly probe the properties of radiation-sources. Choosing the right beam shape of a radio antenna is found crucial for this analysis. Square Kilometre Array (SKA) is found to be a suitable apparatus for this analysis in terms of sensitivity, even though some deterioration of the data for this purpose is unavoidable under the planned size of the antenna core.