On a New Approach to Total Domination Polynomials

For a graph G = (V,E), the open neighbourhood hypergraph of G, denoted by ONH(G), is the hypergraph with vertex set V and edge set {NG(x)|x ∈ V }. A vertex cover in ONH(G) is a set of vertices intersecting every edge of ONH(G), which is equivalent to a total dominating set in G. In this paper, we present a novel approach to determining the total domination polynomials of certain classes of graphs by employing the concept of vertex covering sets. Total domination polynomials, which encode the number of total dominating sets of various cardinalities, offer valuable insight into the structural properties of graphs. Our method establishes an easy and systematic relationship between vertex covers and total dominating sets, allowing for a new algebraic formulation of the total domination polynomial. We apply this technique to specific families of graphs and express the total domination polynomials in terms of vertex cover polynomials. This approach not only simplifies the computation of total domination polynomials for these classes but also highlights deeper connections between domination and covering parameters in graph theory. The results contribute to both the theoretical understanding and practical computation of domination-based graph invariants.