In cove-edged zigzag graphene nanoribbons (ZGNR-Cs), one terminal CH group per length unit is removed on each zigzag edge, forming a regular pattern of coves that controls their electronic structure. Based on three structural parameters that unambiguously characterize the atomistic structure of ZGNR-Cs, we present a scheme that classifies their electronic state (i.e., if they are metallic, topological insulators, or trivial semiconductors) for all possible widths N, unit lengths a, and cove position offsets at both edges b, thus showing the direct structure-electronic structure relation. We further present an empirical formula to estimate the band gap of the semiconducting ribbons from N, a, and b. Finally, we identify all geometrically possible ribbon terminations and provide rules to construct ZGNR-Cs with a well-defined electronic structure.