A Linear Kernel for Independent Set Reconfiguration in Planar Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908392328527872 |
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| author | Bousquet, Nicolas Cranston, Daniel W. |
| author_facet | Bousquet, Nicolas Cranston, Daniel W. |
| contents | Fix a positive integer $r$, and a graph $G$ that is $K_{3,r}$-minor-free. Let $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ into $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kamiński, and Ono (2014) to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$.
Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a $K_{3,r}$-minor-free graph into an equivalent problem on a ($K_{3,r}$-minor-free) graph with order $O(k)$. This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz (2024) and improves the recent quadratic kernel of Cranston, Mühlenthaler, and Peyrille (2024+). For planar graphs, we further strengthen this upper bound to get a kernel of size at most $42k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Linear Kernel for Independent Set Reconfiguration in Planar Graphs Bousquet, Nicolas Cranston, Daniel W. Combinatorics Computational Complexity 05C69, 05C10, 05C85, 68R10 Fix a positive integer $r$, and a graph $G$ that is $K_{3,r}$-minor-free. Let $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ into $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kamiński, and Ono (2014) to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a $K_{3,r}$-minor-free graph into an equivalent problem on a ($K_{3,r}$-minor-free) graph with order $O(k)$. This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz (2024) and improves the recent quadratic kernel of Cranston, Mühlenthaler, and Peyrille (2024+). For planar graphs, we further strengthen this upper bound to get a kernel of size at most $42k$. |
| title | A Linear Kernel for Independent Set Reconfiguration in Planar Graphs |
| topic | Combinatorics Computational Complexity 05C69, 05C10, 05C85, 68R10 |
| url | https://arxiv.org/abs/2506.03319 |