On Fractional Fragility Rates of Graph Classes
Sur les taux de fragilité fractionnaire des classes de graphes
Résumé
We consider, for every positive integer a, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with probability at most 1/a. Among other results, we prove that for every positive integer a and every planar graph G, there exists such a probability distribution with the additional property that for any set X in the support of the distribution, the graph G − X has component-size at most (∆(G) − 1) a+O(√ a) , or treedepth at most O(a 3 log 2 (a)). We also provide nearly-matching lower bounds.
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