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 A preliminary version of this paper appeared in

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3 2 .

A graph G is equimatchable if each matching in G is a subset of a maximum-size matching and it is factor critical if G − v has a perfect matching for each vertex v of G. It is known that any 2-connected equimatchable graph is either bipartite or factor critical. We prove that for 2connected factor-critical equimatchable graph G the graph G \ (V (M) ∪ {v}) is either K 2n or K n,n for some n for any vertex v of G and any minimal matching M such that {v} is a component of G \ V (M). We use this result to improve the upper bounds on the maximum number of vertices of 2-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus g to 4 √ g + 17 if g ≤ 2 and to 12 √ g + 5 if g ≥ 3. Moreover, for any nonnegative integer g we construct a 2-connected equimatchable factor-critical graph with genus g and more than 4 √ 2g vertices, which establishes that the maximum size of such graphs is ( √ g).

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