Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Token sliding on graphs of girth five   In the Token Sliding problem we are given a graph $G$ and two independent
sets $I_s$ and $I_t$ in $G$ of size $k \geq 1$. The goal is to decide whether
there exists a sequence $\langle I_1, I_2, \ldots, I_\ell \rangle$ of
independent sets such that for all $i \in \{1,\ldots, \ell\}$ the set $I_i$ is
an independent set of size $k$, $I_1 = I_s$, $I_\ell = I_t$ and $I_i \triangle
I_{i + 1} = \{u, v\} \in E(G)$. Intuitively, we view each independent set as a
collection of tokens placed on the vertices of the graph. Then, the problem
asks whether there exists a sequence of independent sets that transforms $I_s$
into $I_t$ where at each step we are allowed to slide one token from a vertex
to a neighboring vertex. In this paper, we focus on the parameterized
complexity of Token Sliding parameterized by $k$. As shown by Bartier et al.,
the problem is W[1]-hard on graphs of girth four or less, and the authors posed
the question of whether there exists a constant $p \geq 5$ such that the
problem becomes fixed-parameter tractable on graphs of girth at least $p$. We
answer their question positively and prove that the problem is indeed
fixed-parameter tractable on graphs of girth five or more, which establishes a
full classification of the tractability of Token Sliding parameterized by the
number of tokens based on the girth of the input graph.

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The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
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The categories of this paper are: eess
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys