Abstract
In this paper, selected domination parameters are discussed and proved especially after expanding the graph by duplicating vertices or edges. The tadpole domination number after expansion is obtained in terms of the old tadpole domination number and the maximal path of the original graph, tadpole domination number was determined for any graph after the duplication of the order of the vertices set of G or the duplication of the size of the edges set. Determining if a graph is Hamiltonian is much more problematic. Therefore, the leading outcome in this paper is that we provide that if an expanded graph (by duplicating each vertex by an edge) has tadpole domination, then the original (initial) graph has Hamiltonian path and vice versa.
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