Abstract
An Italian dominating function on a graph G with vertex set V(G) is a function \(f :V(G) \rightarrow \{0,1,2\}\) having the property that for every vertex v with \(f(v) = 0\), at least two neighbors of v are assigned 1 under f or at least one neighbor of v is assigned 2 under f. The weight of an Italian dominating function f is the sum of the values assigned to all the vertices under f. The Italian domination number of G, denoted by \(\gamma _{I}(G)\), is the minimum weight of an Italian dominating of G. It is known that if G is a connected graph of order \(n \ge 3\), then \(\gamma _{I}(G) \le \frac{3}{4}n\). Further, if G has minimum degree at least 2, then \(\gamma _{I}(G) \le \frac{2}{3}n\). In this paper, we characterize the connected graphs achieving equality in these bounds. In addition, we prove Nordhaus–Gaddum inequalities for the Italian domination number.
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The research of the first and second authors was supported in part by the University of Johannesburg.
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Communicated by Xueliang Li.
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Haynes, T.W., Henning, M.A. & Volkmann, L. Graphs with Large Italian Domination Number. Bull. Malays. Math. Sci. Soc. 43, 4273–4287 (2020). https://doi.org/10.1007/s40840-020-00921-y
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DOI: https://doi.org/10.1007/s40840-020-00921-y