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Dominating sets and domination polynomials of paths. (English) Zbl 1177.05081
Summary: Let G=(V,E) be a simple graph. A set SV is a dominating set of G, if every vertex in VS is adjacent to at least one vertex in S. Let 𝒫 n i be the family of all dominating sets of a path P n with cardinality i, and let d(P n ,j)=|𝒫 n j |. In this paper, we construct 𝒫 n i , and obtain a recursive formula for d(P n ,i). Using this recursive formula, we consider the polynomial D(P n ,x)= i=n/3 n d(P n ,i)x i , which we call domination polynomial of paths and obtain some properties of this polynomial.
05C69Dominating sets, independent sets, cliques
05C31Graph polynomials
05C38Paths; cycles