Mixed Roman domination and 2-independence in trees

Document Type : Original paper

Author

Sirjan University of Technology, Sirjan 78137, Iran

Abstract

‎‎Let G=(V,E) be a simple graph with vertex set V and edge set E‎. ‎A em mixed Roman dominating function (MRDF) of G is a function f:VE{0,1,2} satisfying the condition that every element xinVE for which f(x)=0 is adjacent‎ ‎or incident to at least one element yVE for which f(y)=2‎. ‎The weight of an‎ ‎MRDF f is xVEf(x)‎. ‎The mixed Roman domination number γR(G) of G is‎ ‎the minimum weight among all mixed Roman dominating functions of G‎. ‎A subset S of V is a 2-independent set of G if every vertex of S has at most one neighbor in S‎. ‎The minimum cardinality of a 2-independent set of G is the 2-independence number β2(G)‎. ‎These two parameters are incomparable in general‎, ‎however‎, ‎we show that if T is a tree‎, ‎then 43β2(T)γR(T) and we characterize all trees attaining the equality‎.

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