  This paper is devoted to the online dominating set problem and its variants.
We believe the paper represents the first systematic study of the effect of two
limitations of online algorithms: making irrevocable decisions while not
knowing the future, and being incremental, i.e., having to maintain solutions
to all prefixes of the input. This is quantified through competitive analyses
of online algorithms against two optimal algorithms, both knowing the entire
input, but only one having to be incremental. We also consider the competitive
ratio of the weaker of the two optimal algorithms against the other.
  We consider important graph classes, distinguishing between connected and not
necessarily connected graphs. For the classic graph classes of trees,
bipartite, planar, and general graphs, we obtain tight results in almost all
cases. We also derive upper and lower bounds for the class of bounded-degree
graphs. From these analyses, we get detailed information regarding the
significance of the necessary requirement that online algorithms be
incremental. In some cases, having to be incremental fully accounts for the
online algorithm's disadvantage.
