H-forming sets in graphs

  • Authors:

  • Teresa W. Haynes;Stephen T. Hedetniemi;Michael A. Henning;Peter J. Slater

  • Affiliations:

  • Department of Mathematics, East Tennessee State University, Box 70663, Johnson City, TN;Department of Computer Science, Clemson University, Clemson, SC;School of Mathematics, Statistics and Information Technology, University of Natal, Private Bay XO1, Pietermaritzburg 3209, South Africa;Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2003

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Abstract

For graphs G and H, a set S ⊆ V(G) is an H-forming set of G if for every v ∈ V(G)-S, there exists a subset R ⊆ S, where |R| = |V(H)|-1, such that the subgraph induced by R ∪ {v} contains H as a subgraph (not necessarily induced). The minimum cardinality of an H-forming set of G is the H-forming number γ{H}(G). The H-forming number of G is a generalization of the domination number γ(G) because γ(G)= γ{P2}(G). We show that γ(G) ≤ γ{P3}(G) ≤ γt(G), where γt(G) is the total domination number of G. For a nontrivial tree T, we show that γ{P3}(T)= γt(T). We also define independent P3-forming sets, give complexity results for the independent P3-forming problem, and characterize the trees having an independent P3-forming set.