On the number of edges in graphs with a given weakly connected domination number

  • Authors:

  • Laura A. Sanchis

  • Affiliations:

  • Department of Computer Science, Colgate University, Hamilton, NY

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

A dominating set for a graph G = (V, E) is a subset of vertices V' ⊆ V such that for all υ ∈ V - V' there exists some u ∈ V' adjacent to υ. The domination number of G, denoted by λ(G), is the size of its smallest dominating set. A dominating set is weakly connected if the edges not incident on any vertices of the dominating set do not separate the graph (Discrete Math. 167-168 (1997) 261). The weakly connected domination number of G is the size of its smallest weakly connected dominating set. We show in this paper that the maximum number of edges that a graph with n vertices and weakly connected domination number equal to d ≥ 3 can have is (n-d-12). We also characterize the extremal graphs attaining this bound.