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Abstract
Let G = (V, E) be a simple Graph. A set S V(G) is a vertex-edge dominating set (or simply ve-dominating set) if for all edges e E(G), there exist a vertex v S such that v dominates e. In this paper, we study the concept of vertex-edge domination polynomial of the cycle Cn. The vertex-edge domination polynomial of Cn is Dve(Cn, x) = |V(G)| n 4 i dve(Cn, i)xi, where dve(Cn, i) is the number of vertex-edge dominating sets of Cn with cardinality i. We obtain some properties of Dve(Cn, x) and its co-efficients. Also, we calculate the recursive formula to derive the vertex-edge domination polynomials of cycles.
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