Strong hub cover pebbling number

Abstract: In a graph $G$, we define a set of vertices to be a \emph{strong hub set} if for any two vertices in $G$, we can find a path between them whose internal vertices are all in this set. We define the \emph{strong hub cover pebbling number} of $G$, denoted by $h_s^*(G)$, to be the smallest $t$ such that for any initial configuration with $t$ pebbles on $G$, we can make some pebbling moves (a pebbling move consists of removing two pebbles from a vertex $v$ and adding one pebble to another vertex adjacent to $v$) so that there is a strong hub set with every vertex in it having a pebble. We determine the strong hub cover pebbling numbers of paths, stars, and books.