On TD Polynomials of Ring Sum of Graphs

Total domination polynomials enumerate the total dominating sets of a graph according to their cardinalities and provide an algebraic means of examining selected graph constructions. This study investigates total domination polynomials for certain graphs obtained through the ring sum, or symmetric difference, operation. The graph families considered are formed from path graphs and cycle graphs together with star graphs. The analysis employs vertex-cover polynomials and the correspondence between total dominating sets of a graph and vertex covering sets of its open-neighbourhood hypergraph. After establishing the relevant definitions and vertex-cover identities, the study derives expressions for the total domination polynomials of ring sums involving odd and even cycles with stars. It then examines ring sums of paths with stars, considering the parity of the path order and the position of the vertex shared with the star. The derivations use decompositions of open-neighbourhood hypergraphs and products of vertex-cover polynomials to obtain the stated expressions. The resulting formulae represent the total domination polynomials of the considered graph families in terms of vertex-cover polynomials of paths and factors arising from the pendant vertices of the star graphs. Within the specified configurations, the study illustrates the use of established vertex-cover methods for analysing total domination in graphs formed through ring sum operations.