2023/08/18 by Alikhani, Saeid, Aghaei, Fatemeh
#05C99 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.11028
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 characterization of graphs G with π2^*(G)=5 stated as an open problem. In this paper, we solve this problem. We improve the upper bound of the 2-restricted optimal pebbling number of trees of order n. Also, we study 2-restricted optimal pebbling number of some grid graphs, corona and neighborhood corona of two specific graphs.