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 $$\Pi $$ Π at all times. We consider only hereditary properties $$\Pi $$ Π , for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs $${{\mathcal{F}}}$$ F and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the Delayed Connected$${{\mathcal{F}}}$$ F -Node-Deletion Problem, where all graphs of the family $${\mathcal{F}}$$ F have to be connected and almost tight lower and upper bounds for the Delayed$$H$$ H -Node-Deletion Problem, where there is one forbidden induced subgraph H that may be connected or not. For the Delayed$$H$$ H -Node-Deletion Problem the advice complexity is basically an easy function of the size of the biggest component in H. Additionally, we give tight bounds on the Delayed Connected$${\mathcal{F}}$$ F -Edge-Deletion Problem, 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 $${\mathcal{F}}$$ F . We give a separate analysis for the Delayed Connected$$H$$ H -Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.

On Betti numbers of flag complexes with forbidden induced subgraphs

We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

Requiring adjacent chords in cycles

Define a new class of graphs by cycles of length 5 or more always having adjacent chords. This is equivalent to cycles of length 5 or more always having noncrossing chords, which is a property that has a known forbidden induced subgraph characterization. Another characterization comes from viewing the graphs in this class in contrast to distance-hereditary graphs (which are characterized by cycles of length 5 or more always having crossing chords). Moreover, the graphs in the new class are those in which every edge of every cycle [Formula: see text] of length 5 or more forms a triangle with a third vertex of [Formula: see text] (generalizing that a graph is chordal if and only if every edge of every cycle [Formula: see text] of length 4 or more forms a triangle with a third vertex of [Formula: see text]). This leads to a strategically-required subgraph characterization of the new class.