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, $C_t$ [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 $tC_3 \uplus C_4$ (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.

翻译：Dallard、Milani\v{c} 与 \v{S}torgel [arXiv '22] 提出：对于所有排除某个固定平面图 $H$ 作为诱导子式的图类，最大独立集问题是否可在多项式时间内求解？他们已证明当 $H$ 为任意平面完全二分图、5-顶点团减去一边或减去两条不相交边时结论成立。若该问题获证，将是对当前研究前沿的深远推广——目前我们甚至不清楚当 $H$ 为7-顶点路径时是否存在多项式时间算法。若将可解性放宽至拟多项式时间算法，我们已知更多成果。事实上，近年来已针对 $t$-顶点环 $C_t$ [Gartland 等, STOC '21] 及 $t$ 个三角形的并集 $tC_3$ [Bonamy 等, SODA '23] 获得拟多项式时间算法。本文对任意整数 $t$ 给出：当 $H$ 为友谊图 $K_1 + tK_2$（$t$ 条不相交边加上一个与它们全相邻的顶点）时，存在运行时间为 $n^{O(t^5)}$ 的多项式时间算法；当 $H$ 为 $tC_3 \uplus C_4$（$t$ 个三角形与一个4-顶点环的不相交并）时，存在运行时间为 $n^{O(t^2 \log n)+t^{O(1)}}$ 的拟多项式时间算法。前者推广了关于排除 $tK_2$ 作为诱导子图的经典结果 [Alekseev, DAM '07]，后者则扩展了 Bonamy 等人的工作。