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 $d(n)=\frac1{10}(n^2+5n+c)$ and $c$ is an explicit constant that depends on $n \bmod 5$. This settles a conjecture by Lea\~nos and the authors, who previously proved the upper bound. The lower bound is computer-assisted and follows the method used by Gon\c{c}alves, Pinlou, Rao and Thomass\'e for grid graphs.