Monophonic rubbling number of some standard graphs

Let G be a connected graph with vertices V and edges E. Rubbling is a recent development in graph theory and combinatorics. In graph rubbling an extra shift is allowed that adds a pebble at a node after the deletion of one pebble each at two neighbouring vertices. For the first time, we introduce the concept of monophonic rubbling numbers into the literature. A monophonic rubbling number, $$ , is the least number m required to ensure that any vertex is reachable from any pebble placement of m pebbles using a monophonic path by a sequence of rubbling shifts. We calculate the upper bound and lower bound using the monophonic rubbling number of some standard graphs and derived graphs.