Maximum Independent Set when excluding an induced minor: K_1 + tK_2 and tC_3 ⊎ C_4
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Dallard, Milanič, and Štorgel [arXiv '22] ask if for every class excluding a fixed planar graph H as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when H is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when H is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the t-vertex cycle, C_t [Gartland et al., STOC '21] and the disjoint union of t triangles, tC_3 [Bonamy et al., SODA '23]. We give, for every integer t, a polynomial-time algorithm running in n^O(t^5) when H is the friendship graph K_1 + tK_2 (t disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in n^O(t^2 log n)+t^O(1) when H is tC_3 ⊎ C_4 (the disjoint union of t triangles and a 4-vertex cycle). The former extends a classical result on graphs excluding tK_2 as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.'s result.
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