Abstract
A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow \{0,1,2,3\}\) such that (i) every vertex v with \(f(v)=0\) is adjacent to at least two vertices assigned a 2 or to at least one vertex assigned a 3, (ii) every vertex v with \(f(v)=1\) is adjacent to at least one vertex w with \(f(w)\ge 2.\) The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number \(\gamma _{\rm dR}(G)\) equals the minimum weight of a double Roman dominating function on G. Beeler, Haynes and Hedetniemi showed that for every non-trivial tree T, \(\gamma _{\rm dR}(T)\ge 2\gamma (T)+1,\) where \(\gamma (T)\) is the domination number of T. A characterization of extremal trees attaining this bound was given by three of us. In this paper, we characterize all trees T with \(\gamma _{\rm dR}(T)=2\gamma (T)+2\).
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Ahangar, H.A., Amjadi, J., Chellali, M. et al. Trees with Double Roman Domination Number Twice the Domination Number Plus Two. Iran J Sci Technol Trans Sci 43, 1081–1088 (2019). https://doi.org/10.1007/s40995-018-0535-7
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DOI: https://doi.org/10.1007/s40995-018-0535-7