On the complexity of some hop domination parameters

Nader Jafari Rad, Elahe Shabani


A hop Roman dominating function (HRDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} having the property that for every vertex v ∈ V with f(v) = 0 there is a vertex u with f(u) = 2 and d(u, v) = 2. The weight of an HRDF f is the sum of its values on V. The minimum weight of an HRDF on G is called the hop Roman domination number of G. An HRDF f is a hop Roman independent dominating function (HRIDF) if for any pair v, w with f(v) > 0 and f(w) > 0, d(v, w) ≠ 2. The minimum weight of an HRIDF on G is called the hop Roman independent domination number of G. In this paper, we study the complexity of the hop independent dominating problem, the hop Roman domination function problem and the hop Roman independent domination function problem, and show that the decision problem for each of the above three problems is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.


dominating set, hop dominating set, hop independent set, hop Roman dominating function, hop Roman independent dominating function, complexity

Full Text:


DOI: http://dx.doi.org/10.5614/ejgta.2019.7.1.6


S.K. Ayyaswamy, B. Krishnakumari, C. Natarajan and Y. B. Venkatakrishnan, Bounds on the hop domination number of a tree, Proceedings of Math. Sci., Indian Academy of Science (2015), DOI:10.1007/s12044-015-0251-6.

S.K. Ayyaswamy, C. Natarajan, Hop domination in Graphs, Discussiones Math. Graph The- ory (Submitted).

E.J. Cockayane, P.M. Dreyer Jr., S.M. Hedetniemi and S.T. Hedetniemi, On Roman domination in graphs, Discrete Math. 278 (2004), 11–22.

M. Farhadi Jalalvand and N. Jafari Rad, On the complexity of k-step and k-hop dominating sets in graphs, Math. Montisnigri 40 (2017), 36–41.

M.R. Garey, D.S. Johnson, Computers and Intractibility: A Guide to the Theory of NP-completeness, Freeman, New York, 1979.

T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York, 1998.

M.A. Henning and N. Jarfari Rad, On 2-step and hop dominating sets in graphs, Graphs Combin. 33 (4) (2017), 913–927.

R.M. Karp, Reducibility among combinatorial problems, In R.E. Miller and J.W. Thatcher (editors). Complexity of Computer Computations. New York: Plenum. 85–103.

C. Natarajan and S.K. Ayyaswamy, Hop domination in graphs-II, An. Stt. Univ. Ovidius Constanta 23 (2) (2015), 187–199.

C.S. ReVelle, K.E. Rosing, Defendens imperium Romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (2000), 585–594.

E. Shabani, Hop Roman domination in graphs, Manuscript (2017).

I. Stewart, Defend the roman empire!, Sci. Amer. 281 (6) (1999), 136–139.


  • There are currently no refbacks.

ISSN: 2338-2287

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View EJGTA Stats