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On the Roman Domination Number of Graphs

Qajar, Sahar | 2013

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 45059 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. Let G be a graph. A labeling f : V (G) ! f0; 1; 2g is called a Roman dominating function, if every vertex u with f(u) = 0 has at least a neighbor v with f(v) = 2. Define the weight of a Roman dominating function f to be w(f) =Σv2V (G) f(v). The Roman domination number of G is R(G) = minfw(f) : f is a Roman dominating functiong. Some other parameters are defined based on Roman domination number. A Roman bondage number bR(G) of G is the minimum cardinality of all sets E E(G) for which R(G E) > R(G). The edge Roman domination number of G, LR(G), is defined as R(L(G)), where L(G) is the line graph of G. In this thesis, after determining the exact value of the Roman bondage number for some family of graphs, we introduce a sharp upper ound for general graphs. Moreover, for some families of planar graphs we improve the upper bounds. More precisely, we prove that 15 is an upper bound for the Roman bondage number of planar graphs. We introduce some new upper bounds for the edge Roman domination number in some families of graphs, as well
  9. Keywords:
  10. Roman Domination Number ; Edge Roman Domination Number ; Roman Bondage Number