Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

[2]  Young Soo Kwon,et al.  Perfect domination sets in Cayley graphs , 2014, Discret. Appl. Math..

[3]  Hadi Rahbani,et al.  On perfect Roman domination number in trees: complexity and bounds , 2019, J. Comb. Optim..

[4]  Quentin F. Stout,et al.  PERFECT DOMINATING SETS , 1990 .

[5]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[6]  Italo J. Dejter.  Perfect domination in regular grid graphs , 2008, Australas. J Comb..

[7]  Douglas F. Rall,et al.  Associative graph products and their independence, domination and coloring numbers , 1996, Discuss. Math. Graph Theory.

[8]  Tadeja Kraner Sumenjak,et al.  Rainbow domination in the lexicographic product of graphs , 2012, Discret. Appl. Math..

[9]  Benjier H. Arriola,et al.  Doubly connected domination in the corona and lexicographic product of graphs , 2014 .

[10]  Dorota Kuziak,et al.  Total Roman domination in the lexicographic product of graphs , 2019, Discret. Appl. Math..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  J. A. Rodríguez-Velázquez,et al.  From Italian domination in lexicographic product graphs to w-domination in graphs , 2020, Ars Math. Contemp..

[13]  G. Nemhauser,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2014 .

[14]  Juan A. Rodríguez-Velázquez,et al.  On the weak Roman domination number of lexicographic product graphs , 2019, Discret. Appl. Math..

[15]  Ismael González Yero,et al.  Efficient open domination in graph products , 2014, Discret. Math. Theor. Comput. Sci..

[16]  Jiamei Song,et al.  Note on the perfect Roman domination number of graphs , 2020, Appl. Math. Comput..

[17]  T. Haynes.  Domination in Graphs : Volume 2: Advanced Topics , 2017 .

[18]  Juan A. Rodríguez-Velázquez,et al.  On the super domination number of lexicographic product graphs , 2019, Discret. Appl. Math..

[19]  William F. Klostermeyer.  A Taxonomy of Perfect Domination , 2015 .

[20]  P. Roushini Leely Pushpam,et al.  On Total Roman Domination in Graphs , 2016, ICTCSDM.

[21]  Gary MacGillivray,et al.  Perfect Roman domination in trees , 2018, Discret. Appl. Math..

[22]  Tadeja Kraner Sumenjak,et al.  On the Roman domination in the lexicographic product of graphs , 2012, Discret. Appl. Math..

[23]  J. Mark Keil,et al.  Perfect Roman domination in graphs , 2019, Theor. Comput. Sci..

[24]  Stephen T. Hedetniemi,et al.  Roman domination in graphs , 2004, Discret. Math..

[25]  Michael R. Fellows,et al.  Perfect domination , 1991, Australas. J Comb..

[26]  Xindong Zhang,et al.  Domination in lexicographic product graphs , 2011, Ars Comb..

[27]  J. Moon,et al.  Relations between packing and covering numbers of a tree , 1975 .

[28]  Juan A. Rodríguez-Velázquez,et al.  Double domination in lexicographic product graphs , 2020, Discret. Appl. Math..

[29]  Seyed Mahmoud Sheikholeslami,et al.  A characterization of perfect Roman trees , 2020, Discret. Appl. Math..

[30]  V. Yegnanarayanan,et al.  On product graphs , 2012 .