June 10, 2023
Doost Ali Mojdeh

Doost Ali Mojdeh

Degree: Professor
Address: Department of Mathematics, University of Mazandaran, Babolsar, Iran
Education: Ph.D in Mathematics (Graph Theory and Combinatorics)
Phone: 011-35302448
Faculty: Faculty of Mathematical Sciences


Title Outer independent rainbow dominating functions in graphs
Type Article
outer-independent rainbow domination, $K_{1,r}-free graphs, trees
Journal Opuscula Mathematica
DOI https://doi.org/10.7494/OpMath.2020.40.5.599
Researchers Zhila Mansouri (First researcher) , Doost Ali Mojdeh (Second researcher)


A 2-rainbow dominating function (2-rD function) of a graph G=(V,E) is a function f:V(G)→{∅,{1},{2},{1,2}} having the property that if f(x)=∅, then f(N(x))={1,2}. The 2-rainbow domination number γr2(G) is the minimum weight of ∑v∈V(G)|f(v)| taken over all 2-rainbow dominating functions f. An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph G is a 2-rD function f for which the set of all v∈V(G) with f(v)=∅ is independent. The outer independent 2-rainbow domination number γoir2(G) is the minimum weight of an OI2-rD function of G. In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on γoir2(G). Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair (a,b) is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if a+1≤b≤2a.