On 2-restricted optimal pebbling number of a graph

Let G=(V,E) be a simple graph. A function f:V→N∪{0} is called a configuration of pebbles on the vertices of G and the weight of f is w(f)=∑u∈Vf(u) which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex u to one of its neighbors v reduces f(u) by two and increases f(v) by one. A pebbling configuration f is said to be solvable if for every vertex v, there exists a sequence (possibly empty) of pebbling moves that results in a pebble on v. A pebbling configuration f is a t-restricted pebbling configuration (abbreviated tRPC) if f(v)≤t for all v∈V. The t-restricted optimal pebbling number πt∗(G) is the minimum weight of a solvable tRPC on G. Chellali et.al. [Discrete Appl. Math. 221 (2017) 46–53] characterized connected graphs G having small 2-restricted optimal pebbling numbers and the characterization of graphs G with π2∗(G)=5 stated as an open problem. In this paper, we solve this problem. We present an improved upper bound for 2-restricted optimal pebbling number and characterize trees which attain the bound. Also we study the 2-restricted optimal pebbling number of some grid graphs, corona and neighborhood corona of two specific graphs.