Maximum Independent Set when excluding an induced minor: $K_1 + tK_2$ and $tC_3 \uplus C_4$

TL;DR

This work advances the study of Maximum Independent Set under induced-minor exclusions for planar graphs by providing a polynomial-time algorithm for the friendship graph $K_1+tK_2$-induced-minor-free graphs and a quasipolynomial-time algorithm for the disjoint union $tC_3 ⊔ C_4$-induced-minor-free graphs. The polynomial result relies on a layered BFS-based decomposition, bounding adhesion with path-decompositions, and a DP over components after removing a carefully chosen $tK_2$-subgraph, achieving $n^{O(t^5)}$ time. The quasipolynomial result uses a hole-elimination branching rule and a reduction to chordal cores via a structured collection of triangles, yielding a complexity of $n^{O(t^2 \log n) + f(t)}$ with $f(t)$ single-exponential. Together, these contributions extend prior results (Alekseev; Bonamy et al.) and illuminate new structural avenues for MIS in induced-minor-free graph classes, while leaving open several natural extensions to other planar patterns.

Abstract

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 $n^{O(t^2 \log n)+f(t)}$, with $f$ a single-exponential function, 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.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Proposition 5
• Lemma 9