Faster parameterized algorithms for modification problems to minor-closed classes

TL;DR

The paper tackles two core graph-modification problems—Vertex Deletion to a minor-closed class ${\cal G}$ and Elimination Distance to ${\cal G}$—and provides explicit, parametric fixed-parameter tractable (FPT) algorithms with clearly delineated dependencies on $k$. It develops a cohesive framework built on flat walls, boundaried graphs, and canonical partitions to enable uniform DP-based solutions, along with precise obstruction-size bounds and treewidth-based reductions. A major contribution is the first explicit parametric dependence for Elimination Distance to minor-closed classes, including specialized fast treatments when the obstruction set contains apex-graphs. The results unify and extend prior approaches (notably Sau, Stamoulis, Thilikos) and offer practical, constructive bounds that advance understanding of modification problems in minor-closed graph classes.

Abstract

Let ${\cal G}$ be a minor-closed graph class and let $G$ be an $n$-vertex graph. We say that $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. Our first result is an algorithm that decides whether $G$ is a $k$-apex of ${\cal G}$ in time $2^{{\sf poly}(k)}\cdot n^2$, where ${\sf poly}$ is a polynomial function depending on ${\cal G}$. This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020], whose running time was $2^{{\sf poly}(k)}\cdot n^3$. The elimination distance of $G$ to ${\cal G}$, denoted by ${\sf ed}_{\cal G}(G)$, is the minimum number of rounds required to reduce each connected component of $G$ to a graph in ${\cal G}$ by removing one vertex from each connected component in each round. Bulian and Dawar [Algorithmica 2017] provided an FPT-algorithm, with parameter $k$, to decide whether ${\sf ed}_{\cal G}(G)\leq k$. However, its dependence on $k$ is not explicit. We extend the techniques used in the first algorithm to decide whether ${\sf ed}_{\cal G}(G)\leq k$ in time $2^{2^{2^{{\sf poly}(k)}}}\cdot n^2$. This is the first algorithm for this problem with an explicit parametric dependence in $k$. In the special case where ${\cal G}$ excludes some apex-graph as a minor, we give two alternative algorithms, running in time $2^{2^{{\cal O}(k^2\log k)}}\cdot n^2$ and $2^{{\sf poly}(k)}\cdot n^3$ respectively, where $c$ and ${\sf poly}$ depend on ${\cal G}$. As a stepping stone for these algorithms, we provide an algorithm that decides whether ${\sf ed}_{\cal G}(G)\leq k$ in time $2^{{\cal O}({\sf tw}\cdot k+{\sf tw}\log{\sf tw})}\cdot n$, where ${\sf tw}$ is the treewidth of $G$. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs ${\cal E}_k({\cal G})=\{G\mid{\sf ed}_{\cal G}(G)\leq k\}$.

Figures (8)

• Figure 4: Illustration of a flat wall $W$ inside a graph $G$. The edges of $W$ are depicted in orange and the compass of $W$ in $G$ is the union of all parts of $G$ that are drawn in pink cells. The graph in each such pink cell corresponds to a flap of the flat wall.
• Figure 5: A $13$-wall and its six layers, depicted in alternating orange and green. The central vertices of the wall are depicted in red and the corners are depicted in blue.
• Figure 6: A $5$-wall and its canonical partition ${\cal Q}$. The red bag is the external bag $Q_{\rm ext}$.
• Figure 7: An annotated tree made of a rooted tree (left) and a boundaried graph (right). The numbers in the left figure correspond to the pre-images of $f$ for the nodes of $V(T)$ and the numbers on the right figure correspond to the images of $\phi$. The function $h$ that gives a value to each node of the tree is not represented.
• Figure 8: The crop and representation operation applied to the annotated tree of \ref{['@comprehends']}. The unlabeled leaves of the tree are iteratively removed. A representative of each component attached to the boundary of the boundaried graph is kept.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• proposition 3.1: SauST21kapiII, \ref{['@hierarchical']}
• proposition 3.2: BasteST20acom
• proposition 3.3: KawarabayashiTW18anew
• proposition 3.4: SauST21kapiII
• proposition 3.5: SauST21kapiII