We investigate rule dependency graphs and their colorings for characterizing the computation of answer sets of logic programs. To this end, we develop a series of operational characterizations in terms of operators on partial colorings. Our characterizations are expressed as (non-deterministically formed) sequences of colorings, turning an uncolored graph into a totally colored one. This results in an operational framework in which different combinations of operators result in different formal properties. Among others, we identify the basic strategy employed by the noMoRe system and justify its algorithmic approach.