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Aequationes mathematicae

, 78:237 | Cite as

Roman domination subdivision number of graphs

  • M. Atapour
  • S. M. Sheikholeslami
  • Abdollah Khodkar
Article
  • 97 Downloads

Summary.

A Roman dominating function on a graph G = (VE) is a function \(f : V \rightarrow \{0, 1, 2\}\) satisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function is the value \(w(f) = \sum_{v\in V} f(v)\). The Roman domination number of a graph G, denoted by \(_{\gamma R}(G)\), equals the minimum weight of a Roman dominating function on G. The Roman domination subdivision number \(sd_{\gamma R}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number. In this paper, first we establish upper bounds on the Roman domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that \(1 \leq sd_{\gamma R}(G) \leq 3\). Finally, we show that the Roman domination subdivision number of a graph can be arbitrarily large.

Mathematics Subject Classification (2000).

05C69 

Keywords.

Domination in graphs Roman domination number Roman domination subdivision number 

Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  • M. Atapour
    • 1
  • S. M. Sheikholeslami
    • 1
  • Abdollah Khodkar
    • 2
  1. 1.Department of MathematicsAzarbaijan University of Tarbiat MoallemTabrizI. R. Iran
  2. 2.Department of MathematicsUniversity of West GeorgiaCarrolltonU.S.A.

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