CpG islands are genome subsequences with an unexpectedly high number of CG di-nucleotides. They are typically identified using filtering criteria (e.g., G+C% expected vs. observed CpG ratio and length) and are computed using sliding window methods. Most such studies illusively assume an exhaustive search of CpG islands are achieved on the genome sequence of interest. We devise a Lexis diagram and explicitly show that filtering criteria-based definitions of CpG islands are mathematically incomplete and non-operational. These facts imply that the sliding window methods frequently fail to identify a large percentage of subsequences that meet the filtering criteria. We also demonstrate that an exhaustive search is computationally expensive. We develop the Hierarchical Factor Segmentation (HFS) algorithm, a pattern recognition technique with an adaptive model selection device to overcome the incompleteness and non-operational drawbacks, and to achieve effective computations for identifying CpG-islands. The concept of a CpG island "core" is introduced and computed using the HFS algorithm, which is independent from any specific filtering criteria. Upon such a CpG island "core," a CpG-island is constructed using a Lexis diagram. This two-step computational approach provides a nearly exhaustive search for CpG islands that can be practically implemented on whole chromosomes. In a simulation study realistically mimicking CpG-island dynamics through a Hidden Markov Model we demonstrate that this approach retains very high sensitivity and specificity, that is, very low rates of false positives and false negatives. Finally, we apply the HFS algorithm to identify CpG island cores on human chromosome 21.