## 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 ε V:f(v)≠0} induces a subgraph without edges/that is connected/without isolated vertices. For a weight function w{:} V → ℝ, the weight of f is w(f)=Σ _{v ε V} w(v)f(v). The weighted (a,b)-Roman domination number γ^{(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 γ^{(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.

Original language | English (US) |
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Pages (from-to) | 608-619 |

Number of pages | 12 |

Journal | Journal of Combinatorial Optimization |

Volume | 26 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 2013 |

## All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

## Keywords

- Bipartite graphs
- Chordal graphs
- Domination
- Roman domination
- Split graphs
- Strongly chordal graphs