An efficient algorithm for $\mathcal{F}$-subgraph-free Edge Deletion on graphs having a product structure
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| Format: | Preprint |
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2025
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| _version_ | 1866914097878007808 |
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| author | An, Shinwoo Im, Seonghyuk Kim, Seokbeom Lee, Myounghwan |
| author_facet | An, Shinwoo Im, Seonghyuk Kim, Seokbeom Lee, Myounghwan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An efficient algorithm for $\mathcal{F}$-subgraph-free Edge Deletion on graphs having a product structure An, Shinwoo Im, Seonghyuk Kim, Seokbeom Lee, Myounghwan Discrete Mathematics Data Structures and Algorithms Combinatorics 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. |
| title | An efficient algorithm for $\mathcal{F}$-subgraph-free Edge Deletion on graphs having a product structure |
| topic | Discrete Mathematics Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2510.14674 |