Extending Certified Domination: Bondage Numbers in Generalized Petersen Graphs

Abstract

The paper extends the concept of bondage numbers to certified domination, introducing the certified bondage number of a graph. A certified dominating set R is a dominating set of a graph H, if every vertex in R has either zero or at least two neighbours in V\R, where V is the vertex set of H. The minimum cardinality of certified dominating set of H is the certified domination number of H denoted by γcer(H). The bondage number b(H) is defined to be the cardinality of least number of edges F ⊂ E(H) such that γ(H − F) > γ(H). Motivated by this parameter, we extended this concept on certified domination number and defined certified bondage number of a graph H, b+cer(H) [b−cer(H)] to be the cardinality of the least number of edges F ⊂ E(H) such that γcer(H−F) > γcer(H) [γcer(H−F) < γcer(H)] that is minimum number of edges to be removed to increase (or decrease) the certified domination number of H. In this paper, we establish the values of certified bondage number for generalised Petersen graphs P(n, k), where k = 1, 2, as well as for certain classes of graphs.