Semitotal bondage number of certain graphs

Abstract

A set S of vertices in a graph G with no isolated vertices is a semitotal dominating set, abbreviated STDset, of G if S is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, denoted by 2() t G  , is the minimum cardinality of a STD-set of G. We say that a vertex v is semitotal dominated by a set S of vertices in G if v ∈ S and v is within distance 2 from some other vertex of S in G or v ∈ V(G)\S and v is dominated by S in G. The semitotal bondage number 2() t bG of G is the minimum number of edges whose removal enlarges the semitotal domination number. Let G be a graph with no isolated vertices. Then the edge set is a semitotal bondage edge set satisfying the properties: (i) there is no isolated vertex in and (ii) . If at least one semitotal bondage edge set can be found for the graph G, we define semitotal bondage number, denoted by , such that . Otherwise the value of the semitotal bondage number of the graph is . In this paper, we establish lower bound for the semitotal bondage number of a vertex-transitive graph. We also obtain lower bounds for graphs by considering the total bondage number. As applications, we study the semitotal bondage numbers for some certain graphs such as complete, complete bipartite, path, cycle, wheel, star graphs and determine the exact values.