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Vertex domination of generalized Petersen graphs

Javad Ebrahimi, B ; Sharif University of Technology | 2009

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2009.01.018
  3. Publisher: 2009
  4. Abstract:
  5. In a graph G a vertex v dominates all its neighbors and itself. A set D of vertices of G is (vertex) dominating set if each vertex of G is dominated by at least one vertex in D. The (vertex) domination number of G, denoted by γ (G), is the cardinality of a minimum dominating set of G. A set D of vertices in G is efficient dominating set if every vertex of G is dominated by exactly one vertex of D. For natural numbers n and k, where n > 2 k, a generalized Petersen graphP (n, k) is obtained by letting its vertex set be {u1, u2, ..., un} ∪ {v1, v2, ..., vn} and its edge set be the union of {ui ui + 1, ui vi, vi vi + l} over 1 ≤ i ≤ n, where subscripts are reduced modulo n. We prove a necessary and sufficient condition for these graphs to have an efficient dominating set, and we determine exact values of γ (P (n, k)) for k ∈ {1, 2, 3}. Also we prove that for an odd number k, γ (P (n, k)) = frac(n, 2) + O (k) and for an even number k > 2, γ (P (n, k)) ≤ frac(5 n, 9) + O (k). © 2009 Elsevier B.V. All rights reserved
  6. Keywords:
  7. Cardinality ; Dominating sets ; Domination numbers ; Edge-sets ; Efficient dominating sets ; Efficient domination ; Generalized Petersen graph ; Graph g ; Minimum dominating sets ; Natural numbers ; Odd numbers ; Perfect domination ; Sufficient conditions ; Vertex domination ; Vertex sets ; Graph theory
  8. Source: Discrete Mathematics ; Volume 309, Issue 13 , 2009 , Pages 4355-4361 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0012365X0900020X