# 2-Rainbow Domination and Its Practical Variation on Weighted Graphs

Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 20)

## Abstract

Let G(V, E) be a simple graph with n-vertex-set V and m-edge-set E. Two positive weights, w 1 (v) and w 2 (v), are assigned to each vertex v. For each vV, let N(v) = {u | uV and (u, v) ∈ E} and N[v] = {v} ∪ N(v). A 2-rainbow function of G is a function f mapping each vertex v to f(v) = f 2(v) f 1(v), f 2(v), f 1(v) ∈ {0, 1}. The weight of f is defined as $$w(f) = \sum_{v \in V[f_1(v)w_1(v) + f_2(v)w_2(v)]}$$. A 2-rainbow function f of G is called a 2-rainbow dominating function if Θ u ∈ N(v) f(u) = 11, for all vertices v with f(v) = 00, where Θ u ∈ N(v) f(u) is the result of performing bit-wise Boolean OR on f(u), for all uN(v). Our problem is to obtain a 2-rainbow dominating function f of G such that w(f) is minimized. This paper first proves that the problem is NP-hard on unweighted planar graphs. Then, an O(n)-time algorithm for the problem on trees is proposed using the dynamic programming strategy. Finally, a practical variation, called the weighted minimum tuple 2-rainbow domination problem, is proposed and the relationship between it and the weighted minimum domination problem is established.

### Keywords

2-rainbow domination functions tuple 2-rainbow domination functions planar graphs trees NP-hard

## Preview

### References

1. 1.
Ali, M., Rahim, M.T., Zeb, M., Ali, G.: On 2-Rainbow Domination of Some Families of Graphs. International Journal of Mathematics and Soft Computing 1, 47–53 (2011)Google Scholar
2. 2.
Brešar, B., Henning, M.A., Rall, D.F.: Rainbow Domination in Graphs. Taiwanese Journal of Mathematics 12, 213–225 (2008)
3. 3.
Brešar, B., Šumenjak, T.K.: On the 2-Ranbow Domination in Graphs. Discrete Applied Mathematics 155, 2394–2400 (2007)
4. 4.
Chang, G.J., Wu, J., Zhu, X.: Rainbow Domination on Trees. Discrete Applied Mathematics 158, 8–12 (2010)
5. 5.
Chartrand, G., Zhang, P.: Introduction to Graph Theory. McGraw-Hill International Edition (2005)Google Scholar
6. 6.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Bell Laboratories, Freeman & Co., Murray Hill (1979)Google Scholar
7. 7.
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York (1998)
8. 8.
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs. Advanced Topics. Marcel Dekker, Inc., New York (1998)Google Scholar
9. 9.
Lee, R.C.T., Tseng, S.S., Chang, R.C., Tsai, Y.T.: Introduction to the Design and Analysis of Algorithms. McGraw Hill Education, Asia (2005)Google Scholar
10. 10.
Meierling, D., Sheikholeslami, S.M., Volkmann, L.: Nordhaus-Gaddum Bounds on the k-Rainbow Domatic Number of a Graph. Applied Mathematics Letters 24, 1758–1761 (2011)
11. 11.
Tong, C., Lin, X., Yang, Y., Luo, M.: 2-Rainbow Domination of Generalized Petersen Graphs P(n, 2). Discrete Applied Mathematics 157, 1932–1937 (2009)
12. 12.
Yen, W.C.-K., Liu, J.-J., Shih, C.-C.: The Weighted Minimum Tuple 2-Rainbow Domination on Graphs. World Academy of Science, Engineering and Technology 62, 1183–1186 (2012)Google Scholar