## Abstract

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V→{-1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=-1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that γ_{sR}(G) and that γ _{sR}(G)≥(3n-4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that γ _{sR}(G)≥3{n+1} - n - 3, and we characterize the extremal graphs.

Original language | English |
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Pages (from-to) | 241-255 |

Number of pages | 15 |

Journal | Journal of Combinatorial Optimization |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2014 |

## Keywords

- Roman domination
- Signed Roman domination
- Signed domination

## ASJC Scopus subject areas

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