The domination number of the $2$-token graph of path graphs

Abstract

We prove that the domination number of the $2$-token graph of the path $P_n$ is $\gamma (F_2(P_n))=d(n)$ for every $n\ge 13$, where $d(n)=\frac{1}{10}(n^2+5n+c)$ and $c$ is an explicit constant that depends on $n\mathrm{mod}5$. This settles a conjecture by Leaños and the authors, who previously proved the upper bound. The lower bound is computer-assisted and follows the method used by Gonçalves, Pinlou, Rao and Thomassé for grid graphs.

arXiv comment14 pages, 6 figures, 2 Tables. First draft (to be updated). Code development, computations, and grammatical editing were assisted by Anthropic's Claude AI