On monophonic pebbling number

Given a connected graph G and a configuration D of pebbles on V(G), a pebble move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A monophonic path is a longest chordless path between two non-adjacent vertices u and v. The line segment that connects two vertices on a curve is known as a chord. The monophonic distance between u and v is the number of vertices in the longest u–v monophonic path, denoted by dμ(u, v) in G. The monophonic pebbling number (MPN) of G is the least number of pebbles needed to guarantee that, from any distribution of pebbles on a graph G, one pebble can be moved to any specified vertex using monophonic paths through pebbling moves. The monophonic t-pebbling number (MtPN) of G is the least number of pebbles needed to guarantee that, from any distribution of pebbles, t pebbles can be moved to any specified vertex using monophonic paths. In this article, we determine the MPN and MtPN of Dutch windmill graphs, square of cycles, tadpole graphs, lollipop graphs, double star path graphs, and fuse graphs, and we also discuss their t-pebbling versions.