Online Node- and Edge-Deletion Problems with Advice

In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class Π at all times. We consider only hereditary properties Π , for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs ℱ and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the D elayed C onnected ℱ -N ode -D eletion P roblem , where all graphs of the family ℱ have to be connected and almost tight lower and upper bounds for the D elayed H -N ode -D eletion P roblem , where there is one forbidden induced subgraph H that may be connected or not. For the D elayed H -N ode -D eletion P roblem the advice complexity is basically an easy function of the size of the biggest component in H . Additionally, we give tight bounds on the D elayed C onnected ℱ -E dge -D eletion P roblem , where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from ℱ . We give a separate analysis for the D elayed C onnected H -E dge -D eletion P roblem , which is less general but admits a bound that is easier to compute.