Packing chromatic vertex-critical graphs

For a non-decreasing sequence of positive integers $S = (s_1,s_2,\ldots)$, the {\em $S$-packing chromatic number} $\chi_S(G)$ of $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $X_i$, $i \in [k]$, where vertices in $X_i$ are pairwise at distance greater than $s_i$. In this paper we introduce $S$-packing chromatic vertex-critical graphs, $\chi_{S}$-critical for short, as the graphs in which $\chi_{S}(G-u) 1$. We also deal with $k$-$\chi_{S}$-criticality of trees and caterpillars.