Original Research Article

Article volume = 2022 and issue = 2

Pages: 175–180

Article publication Date: November 21, 2022

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Mixed Double Roman Domination Number in Trees

Mina Valinavaz

Department of Mathematics, Azarbaijan Shahid Madani University Tabriz, I.R. Iran.


Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A{\em mixed double Roman dominating function} $(MDRDF)$ of $G$ is a function $f:V\cup E\rightarrow \{0,1,2,3\}$ satisfying the condition every element $x\in V\cup E$ for which $f(x)=0$, is adjacent or incident to at least two elements $y,y\in V\cup E$ for which $f(y)=f(y)=2$ or one element $y""\in V\cup E$ with $f(y"")=3$, and if $f(x)=1$, then element $x\in V\cup E$ must have at least one neighbor $y\in V\cup E$ with $f(y)\ge 2$. The mixed double Roman dominating number of $G$, denoted by $\gamma^*_{dR}(G)$. The weight of a $MDRDF$ $f$ is $w(f)=\sum\limits_{x\in V\cup E}f(x)$. The mixed double Roman domination number of $G$ is the minimum weight of a mixed double Roman dominating function of $G$.


Double Roman dominating function, Double Roman domination number, Mixed double Roman dominating function, Mixed double Roman domination number.

  • [1] H. Abdollahzadeh Ahangar, M. Chellai, S.M. Sheikholeslam, on the double Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2017), 501–517. 1
  • [2] H. Abdollahzadeh Ahangar,Teresa W. Haynes, J.C. Valenzuela-Tripodoro, Mixed Roman Domination in Graphs, Math. Sci. Soc. (2015) 1
  • [3] H. Abdollahzadeh Ahangar,L. Asgarsharghi, S.M. Sheikholeslami, L. Volkmann, Signed mxed Roman domination numbers in graphs, J. Comb. Obtim. (2015) 1
  • [4] M. Valinavaz, On the edge double Roman domination number of graphs, Commun. Combin. Sci (1)(2021),113-119. 1
  • [5] M. Valinavaz, Results on the edge double Roman domination number of a graph, Commun. Combin. Sci (1)(2021),104-112. 1
  • [6] M. Valinavaz, On the edge double Roman domination number of planar graph, Commun. Combin. Sci (1)(2021),97-103. 1
  • [7] R.A. Beeler, T.W. Haynes, and S.T. Hedetniemi, Double Roman Domination, Discrete Apple. Math. 211(2016), 23-29. 1
  • [8] E.J. Cockayne, S. Goodman, S.A. Hedetniemi, linear algorithm for the domination number of a tree,Inf. Process. Lett. 4,(1975) 41–44 1
  • [9] P. Erdös, A. Meir, On total matching numbers and total covering numbers of complementary graphs. Discret. Math. 19,(1977) 229–233 1
  • [10] P. Hatami, An approximation algorithm for the total covering problem. Discuss. Math. Graph Theory 27(3),(2007) 553–558 1
  • [11] J.K. Lan, G.J. Chang, On the mixed domination problem in graphs. Theor. Comput. Sci.476,(2013) 84–93 1
  • [12] Y. Zhao, L. Kang, M.Y. Sohn, The algorithmic complexity of mixed domination in graphs. Theor. Comput. Sci. 412,(2011) 2387–2392. 1
Cite this article as:
  • Mina Valinavaz, Mixed Double Roman Domination Number in Trees, Communications in Combinatorics, Cryptography & Computer Science, 2022(2), PP.175–180, 2022
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