# Three remarks on π_2 graphs

Let k β₯ 1. A graph G is π_π€ if for any k pairwise disjoint independent vertex subsets A_1, β¦, A_k in G, there exist k pairwise disjoint maximum independent sets S_1, β¦, S_k in G such that A_i β S_i for i β [k]. Recognizing π_1 graphs is co-NP-hard, as shown by ChvΓ‘tal and Hartnell (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing π_π€ graphs is co-NP-hard for k β₯ 2. On the positive side, we show that recognizing π_π€ graphs is, for each kβ₯ 2, FPT parameterized by clique-width and by tree-width. Finally, we construct graphs G that are not π_2 such that, for every vertex v in G and every maximal independent set S in G - N[v], the largest independent set in N(v) β S consists of a single vertex, thereby refuting a conjecture of Levit and Tankus.

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