Domination cover rubbling

Optimal pebbling of complete binary trees and a meta-Fibonacci sequence

“…For example, the optimal pebbling number of a graph has been extended in a variety of ways by restricting the capacity of the pebbling configuration or by placing additional requirements on the pebbling configuration; see, for example, [5,21,19,20]. Graph rubbling is a cognate of pebbling; readers who are interested in graph rubbling should consult [2,3,4,1].…”

“…Recall from §1 that the Conolly-Fox sequence {c n } satisfies the recurrence relation (2) with initial conditions c(1) = 1 and c(2) = 2. We will prove the following theorem.…”

Monophonic rubbling number of some standard graphs

“…The vertex v in graph G is reachable if a pebble can be placed on v using a sequence of pebbling and rubbling shifts. In graph pebbling only pebbling shift is permitted whereas in rubbling both pebbling and rubbling shift is permitted [4] . C. Belford et al in [3] computed the rubbling number and optimal rubbling number in some standard graphs.…”

“…Laszlo F. Papp et al computed the lower bounds on both the optimal pebbling and rubbling number using the distance k domination number in [9] . Robert A. Beeler et al computed the domination cover rubbling number of standard graphs such as path, cycle, and trees and found their bounds in [4] . In article [10] the author studied -super magic labelling of zig-zag, linear chains and also the disjoint union of non-isomorphic copies of both chains which motivated us to implement the cycle graph into our study.…”