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A set S  V is a dominating set of G, if every vertex in V− S is adjacent to atleast one vertex S. Let Dg(Gn, i) be the family of geodetic dominating sets of the graph Gn with cardinality 'i'. Let dg(Gn, i) = | Dg(Gn, i)|. In this paper, we obtain a recursive for dg(Gn, i). Using the recursive formula, we construct the polynomial, g(Gn) = n + 1 2     , g(Gn − {2n}) = n + 1 2     which we call geodetic dominating polynomial of Gn and obtain some properties of this polynomial.", "lang": "en" }, "license": { "url": "https://creativecommons.org/licenses/by/4.0/", "start": "2020-01-02", "applies_to": "vor" }, "references": [ { "key": "ref1", "doi": "10.1155/2009/542040", "unstructured": "Alikhani. S and Peng. Y.H, \"Dominating Sets and Domination Polynomials of Paths\", International Journal of Mathematics and Mathematical Sciences, 10, 2009" }, { "key": "ref2", "unstructured": "Alikhani. 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