An efficient algorithm for $\mathcal{F}$-subgraph-free Edge Deletion on graphs having a product structure

TL;DR

The paper tackles the problem of $\mathcal{F}$-subgraph-free Edge Deletion and proves fixed-parameter tractability on planar graphs (and more generally graphs of bounded genus) using a unified product-structure framework. It embeds the input into a strong product $H\boxtimes P$ with $\mathrm{tw}(H)=O(1)$ and then solves the problem in linear time relative to the input size, with running times that are double-exponential in the parameter tuple $(|\mathcal{F}|,k,r,w)$, thus outperforming FO-model checking bounds based on twin-width. The framework extends to disk graphs of bounded local radius, yielding a similar complexity, and it shows tight ETH-based hardness results indicating the necessity of the parameters and the planarity constraint. Together, these results advance a general, scalable approach for $\mathcal{F}$-subgraph-free Edge Deletion across several graph classes via product-structure decompositions, and they introduce new gadget-based hardness constructions for the problem. The work has practical implications for graph editing problems in planar and near-planar settings, where product-structure techniques enable efficient, parameterized decision procedures.

Abstract

Given a family $\mathcal{F}$ of graphs, a graph is \emph{$\mathcal{F}$-subgraph-free} if it has no subgraph isomorphic to a member of $\mathcal{F}$. We present a fixed-parameter linear-time algorithm that decides whether a planar graph can be made $\mathcal{F}$-subgraph-free by deleting at most $k$ vertices or $k$ edges, where the parameters are $k$, $\lvert \mathcal{F} \rvert$, and the maximum number of vertices in a member of $\mathcal{F}$. The running time of our algorithm is double-exponential in the parameters, which is faster than the algorithm obtained by applying the first-order model checking result for graphs of bounded twin-width. To obtain this result, we develop a unified framework for designing algorithms for this problem on graphs with a ``product structure.'' Using this framework, we also design algorithms for other graph classes that generalize planar graphs. Specifically, the problem admits a fixed-parameter linear time algorithm on disk graphs of bounded local radius, and a fixed-parameter almost-linear time algorithm on graphs of bounded genus. Finally, we show that our result gives a tight fixed-parameter algorithm in the following sense: Even when $\mathcal{F}$ consists of a single graph $F$ and the input is restricted to planar graphs, it is unlikely to drop any parameters $k$ and $\lvert V(F) \rvert$ while preserving fixed-parameter tractability, unless the Exponential-Time Hypothesis fails.

Figures (7)

• Figure 1: An illustration of an arrangement graph
• Figure 2: An illustration of $A'_\mathcal{D}$.
• Figure 3: An example of our reduction. Here, the given formula is $\varphi\equiv C_1\land C_2\land C_3$, where $C_1\equiv x_1\lor x_2\lor x_3$, $C_2\equiv x_2\lor \lnot x_3\lor x_4$, and $C_3\equiv x_1\lor x_2\lor \lnot x_4$, and $\mathcal{X}=\set{x_1,x_2,x_3,x_4}$.
• Figure 4: An illustration of $L(g,g_1,g_2)$ (left) and a splitter $H^{\mathrm{sp}}(g, h, h_1, h_2, h_3)$ (right).
• Figure 5: An illustration of a variable gadget.

Theorems & Definitions (37)

• Theorem 1.1: enright2018deleting
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: dujmovic2020planar, improved in ueckerdt2022improved
• Theorem 1.5
• Theorem 1.6: BMO22
• Theorem 1.7
• Corollary 1.8
• Lemma 1.9: bonnet2021twin2
• Proposition 1.10