Rainbow domination on trees

  • Authors:

  • Gerard J. Chang;Jiaojiao Wu;Xuding Zhu

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan and National Center for ...;Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan;National Center for Theoretical Sciences, Taiwan and Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

This paper studies a variation of domination in graphs called rainbow domination. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2,...,k} such that for any vertex v with f(v)=0@? we have @?"u"@?"N"""G"("v")f(u)={1,2,...,k}. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number @c"r"k(G) of a graph G, that is the minimum value of @?"v"@?"V"("G")|f(v)| where f runs over all k-rainbow dominating functions of G. In this paper, we prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the k-rainbow domination problem on trees. For a given tree T, we also determine the smallest k such that @c"r"k(T)=|V(T)|.