Difference between 2-rainbow domination and Roman domination in graphs

  • Authors:

  • Shinya Fujita;Michitaka Furuya

  • Affiliations:

  • Department of Integrated Design Engineering, Maebashi Institute of Technology, 460-1 Kamisadori, Maebashi, Gunma 371-0816, Japan;Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Sinjuku-ku, Tokyo 162-8601, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

A 2-rainbow dominating functionf of a graph G is a function f:V(G)-2^{^1^,^2^} such that, for each vertex v@?V(G) with f(v)=0@?, @?"u"@?"N"""G"("v")f(u)={1,2}. The minimum of @?"v"@?"V"("G")|f(v)| over all such functions is called the 2-rainbow domination number@c"r"2(G). A Roman dominating functiong of a graph G, is a function g:V(G)-{0,1,2} such that, for each vertex v@?V(G) with g(v)=0, v is adjacent to a vertex u with g(u)=2. The minimum of @?"v"@?"V"("G")g(v) over all such functions is called the Roman domination number@c"R(G). Regarding 0@? as 0, these two dominating functions have a common property that the same three integers are used and a vertex having 0 must be adjacent to a vertex having 2. Motivated by this similarity, we study the relationship between @c"R(G) and @c"r"2(G). We also give some sharp upper bounds on these dominating functions. Moreover, one of our results tells us the following general property in connected graphs: any connected graph G of order n=3 contains a bipartite subgraph H=(A,B) such that @d(H)=1 and |A|-|B|=n/5. The bound on |A|-|B| is best possible.