TOTAL DOMINATION POLYNOMIALS OF SOME SPLITTING GRAPHS
A hypergraph is an ordered pair where V is a finite nonempty set called vertices and E is a collection of subsets of V, called hyperedges or simply edges. A subset T of vertices in a hypergraph H is called a vertex cover if T has a nonempty intersection with every edge of H. The vertex covering number of H is the minimum size of a vertex cover in H. Let be the family of vertex covering sets of H with cardinality i and let be the cardinality of The polynomial is defined as vertex cover polynomial of H. For a graph denotes the hypergraph with vertex set V and edge set In this paper, we prove that the total domination polynomial of a connected graph G is the vertex cover polynomial of Using this result, we determine total domination polynomials of splitting graphs of order k of paths and cycles. Moreover, we introduce the terminology of iterated splitting graph of a graph G and determine its total domination polynomials.
total domination, vertex cover, total domination polynomial.