Dallard, Milani\v{c}, and \v{S}torgel [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, Ct [Gartland et al., STOC '21] and the
disjoint union of t triangles, tC3 [Bonamy et al., SODA '23].
We give, for every integer t, a polynomial-time algorithm running in
nO(t5) when H is the friendship graph K1+tK2 (t disjoint edges
plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm
running in nO(t2logn)+tO(1) when H is tC3⊎C4 (the
disjoint union of t triangles and a 4-vertex cycle). The former extends a
classical result on graphs excluding tK2 as an induced subgraph [Alekseev,
DAM '07], while the latter extends Bonamy et al.'s result.Comment: 15 pages, 2 figure