Abstract
Given real numbers b≥a>0, an (a,b)-Roman dominating function of a graph G=(V,E) is a function f:V→{0,a,b} such that every vertex v with f(v)=0 has a neighbor u with f(u)=b. An independent/connected/total (a,b)-Roman dominating function is an (a,b)-Roman dominating function f such that {v∈V:f(v)≠0} induces a subgraph without edges/that is connected/without isolated vertices. For a weight function \(w{:} V\to\Bbb{R}\), the weight of f is w(f)=∑ v∈V w(v)f(v). The weighted (a,b)-Roman domination number \(\gamma^{(a,b)}_{R}(G,w)\) is the minimum weight of an (a,b)-Roman dominating function of G. Similarly, we can define the weighted independent (a,b)-Roman domination number \(\gamma^{(a,b)}_{Ri}(G,w)\). In this paper, we first prove that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/independent (a,b)-Roman domination problems are NP-complete for bipartite graphs. We also show that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/weighted independent (a,b)-Roman domination problems are NP-complete for chordal graphs. We then give linear-time algorithms for the weighted (a,b)-Roman domination problem with b≥a>0, and the weighted independent (a,b)-Roman domination problem with 2a≥b≥a>0 on strongly chordal graphs with a strong elimination ordering provided.
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References
Burger AP, Cockayne EJ, Grundlingh WR, Mynhardt CM, van Vuuren JH, Winterbach W (2004) Finite order domination in graphs. J Comb Math Comb Comput 49:159–175
Chambers EW, Kinnersley B, Prince N, West DB (2009) Extremal problems for Roman domination. SIAM J Discrete Math 23:1575–1586
Chang GJ (2004) The weighted independent domination problem is NP-complete for chordal graphs. Discrete Appl Math 143:351–352
Chang GJ, Nemhauser GL (1984) The k-domination and k-stability problems on sun-free chordal graphs. SIAM J Algebr Discrete Methods 5:332–345
Cockayne EJ, Favaron O, Mynhardt CM (2003) Secure domination, weak Roman domination and forbidden subgraphs. Bull Inst Comb Appl 39:87–100
Cockayne EJ, Dreyer PA Jr, Hedetniemi SM, Hedetniemi ST (2004) Roman domination in graphs. Discrete Math 278:11–22
Cockayne EJ, Grobler PJP, Gründlingh W, Munganga J, van Vuuren JH (2005) Protection of a graph. Util Math 67:19–32
Farber M (1982) Independent domination in chordal graphs. Oper Res Lett 1:134–138
Farber M (1984) Domination, independent domination and duality in strongly chordal graphs. Discrete Appl Math 7:115–130
Fernau H (2008) Roman domination: a parameterized perspective. Int J Comput Math 85(1):25–38
Garey MR, Johnson DS (1979) Computers and intractability. Freeman, San Francisco
Golumbic MC (2004) Algorithmic graph theory and perfect graphs, 2nd edn. Elsevier, Amsterdam
Hedetniemi ST, Henning MA (2003) Defending the Roman Empire—a new strategy. Discrete Math 266:239–251
Henning MA (2002) A characterization of Roman trees. Discuss Math, Graph Theory 22(2):325–334
Henning MA (2003) Defending the Roman Empire from multiple attacks. Discrete Math 271:101–115
Liu C-H, Chang GJ (2012a) Roman domination on 2-connected graphs. SIAM J Discrete Math 26:193–205
Liu C-H, Chang GJ (2012b) Upper bounds on Roman domination numbers of graphs. Discrete Math 312:1386–1391
Liedloff M, Kloks T, Liu J, Peng S-L (2008) Efficient algorithms for Roman domination on some classes of graphs. Discrete Appl Math 156:3400–3415
Prince N (2006) Thresholds for Roman domination, Manuscript
ReVelle CS (1997a) Can you protect the Roman Empire? Johns Hopkins Mag 49(2):40
ReVelle CS (1997b) Test your solution to “Can you protect the Roman Empire”. Johns Hopkins Mag 49(3):70
ReVelle CS, Rosing KE (2000) Defendens Imperium Romanum: a classical problem in military. Am Math Mon 107(7):585–594
Stewart I (1999) Defend the Roman Empire! Sci Am 281(6):136–139
Song X-X, Wang X-F (2006) Roman domination number and domination number of a tree. Chin Q J Math 21(3):358–367
Xing H-M, Chen X, Chen X-G (2006) A note on Roman domination in graphs. Discrete Math 306:3338–3340
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Supported in part by the National Science Council under grant NSC95-2221-E-002-125-MY3.
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Liu, CH., Chang, G.J. Roman domination on strongly chordal graphs. J Comb Optim 26, 608–619 (2013). https://doi.org/10.1007/s10878-012-9482-y
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DOI: https://doi.org/10.1007/s10878-012-9482-y