No abstract

Objectives: To find the secure domination cover pebbling number for the join of two graphs G(p, q) and G ′ (p ′ , q ′ ). Methods: We define Secure domination cover pebbling number, f sd p (G), of a graph G as the minimum number of pebbles that must be placed on V (G) such that, after a sequence of pebbling moves, the set of vertices with pebbles forms a secure dominating set for G. Findings: We found the secure domination cover pebbling number for the join of two graphs G(p, q) and K n . Also, the secure domination cover pebbling number for the join of two graphs G(p, q) and G ′ (p ′ , q ′ ) is determined when the cardinality of the secure dominating set is 2, 3 and 4. A generalization for the secure domination cover pebbling number of path P n is also found.

In this paper, we introduce a new graph invariant called the secure total domination cover pebbling number, a combination of two graph invariants, namely, 'secure total domination' and 'cover pebbling number'. The secure total domination cover pebbling number of a graph G, denoted by f std p (G), is the minimum number of pebbles that are required to place on V (G), such that after a sequence of pebbling moves, the set of vertices with pebbles forms a total secure dominating set under any configuration of pebbles to the vertices of graph G. The secure total domination cover pebbling number for join of two graphs G(p, q) and G ′ (p ′ , q ′ ) is determined. Also, a generalization of secure total domination cover pebbling number for some families of graphs such as complete graph K n , complete bipartite graph K p,q , complete y-partite graph K p 1 ,p 2 ,...,p y and path P n is found.

An edge pebbling move is defined as the removal of two pebbles from one edge and placing one on the adjacent edge. The maximal matching cover pebbling number, fmmcp(G), of a graph G, is the minimum number of pebbles that must be placed on E(G), such that after a sequence of pebbling moves the set of edges with pebbles forms a maximal matching regardless of the initial configuration. In this paper, we find the maximal matching cover pebbling number for variants of hypercube.