Zulfiqar Zaman, M. Kamal Kumar and Saad Salman Ahmad

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 3, Pages 142—150

DOI: 10.7546/nntdm.2018.24.3.142-150

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## Details

### Authors and affiliations

Zulfiqar Zaman

*Department of Mathematics, Higher College of Technology
Muscat, Oman*

M. Kamal Kumar

*Department of Mathematics, Higher College of Technology
Muscat, Oman*

Saad Salman Ahmad

*Department of Mathematics, Higher College of Technology
Muscat, Oman*

### Abstract

Motivated by the article in Scientific American [8], Michael A. Henning and Stephen T. Hedetniemi explored the strategy of defending the Roman Empire. Cockayne defined Roman dominating function (RDF) on a Graph *G = *(*V*,* E*) to be a function *f* : *V *→ {0, 1, 2} satisfying the condition that every vertex *u* for which *f* (*u*) = 0. is adjacent to at least one vertex *v* for which* f* (*v*) = 2. For a real valued function *f* : *V* → *R* the weight of is The Roman Domination Number (RDN) denoted by *γ _{R }*(

*G*) is the minimum weight among all RDF in

*G*. If

*V*–

*D*contains a Roman dominating function

*f*

^{1}:

*V*→ {0, 1, 2}, where

*D*is the set of vertices

*v*for which

*f*(

*v*) > 0. Then

*f*

^{ 1}is called inverse Roman dominating function (IRDF) on a graph

*G*w.r.t.

*f*. The inverse Roman domination number (IRDN) denoted by

*γ*

^{1}

*(*

_{R}*G*) is the minimum weight among all IRDF in

*G*. In this paper we find few results of RDN and IRDN.

### Keywords

- Domination number
- Inverse domination number
- Roman domination number

### 2010 Mathematics Subject Classification

- 05C69

### References

- Berge, C. (1958) Theory of Graphs and Its Applications, Methuen, London.
- Harary, F. (1975) Graph Theory, Addison Wiley, Reading Mass.
- Henning, M. A., & Hedetniemi, S. T. (2003) Defending the Roman Empire – A new strategy, Discrete Mathematics, 266, 239–251.
- Kamal Kumar, M., & Murali, R. (2014) Inverse Roman domination in some classes of graphs. International Journal of Computer Application, 4(4), 219–238.
- Kamal Kumar, M., & Sudershan Reddy, L. (2013) Inverse Roman domination in graphs, Discrete Mathematics Algorithm and Application, 5(3), 1–4.
- Ore, O. (1962) Theory of Graphs. American Mathematical Society Colloquium Publications, 38 (American Mathematical Society, Providence, RI).
- ReVelle. C. S, Rosing. K. E. (2000) Defendens imperium Romanum: A classical problem in military, Strategy, Amer. Math. Monthly, 107(7), 585–594.
- Stewart, I. (1999) Defend the Roman Empire! Scientific American, 281(6), 136–139.

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## Cite this paper

Zaman, Z., Kamal Kumar, M., & Ahmad, S. S. (2018). Roman and inverse Roman domination in graphs. Notes on Number Theory and Discrete Mathematics, 24(3), 142-150, doi: 10.7546/nntdm.2018.24.3.142-150.