An outer-independent double Roman dominating function (OIDRDF) of a graph G is a function (Formula presented.) such that i) every vertex v with (Formula presented.) is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertex v with (Formula presented.) is adjacent to at least one vertex with label greater than 1, and iii) all vertices labeled by 0 are an independent set. The weight of an OIDRDF is the sum of its function values over all vertices. The outer-independent double Roman domination number γoidR (G) is the minimum weight of an OIDRDF on G. It has been shown that for any tree T of order n ≥ 3, γoidR (T) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show that, for any connected graph G of order n with minimum degree at least two in which the set of vertices with degree at least three is independent, γoidR (T) ≤ 4n/3.
CITATION STYLE
Rao, Y., Kosari, S., Sheikholeslami, S. M., Chellali, M., & Kheibari, M. (2021). On the Outer-Independent Double Roman Domination of Graphs. Frontiers in Applied Mathematics and Statistics, 6. https://doi.org/10.3389/fams.2020.559132
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