2-bondage in graphs

Abstract

A 2-dominating set of a graph G=(V, E) is a set D of vertices of G such that every vertex of V(G) ∖ D has at least two neighbours in D. The 2-domination number of a graph G, denoted by γ₂(G), is the minimum cardinality of a 2-dominating set of G. The 2-bondage number of G, denoted by b ₂(G), is the minimum cardinality among all sets of edges E′ ⊆ E such that γ₂(G−E′)>γ₂(G). If for every E′ ⊆ E we have γ₂(G−E′)=γ₂(G), then we define b ₂(G)=0, and we say that G is a γ₂-strongly stable graph. First, we discuss the basic properties of 2-bondage in graphs. We find the 2-bondage numbers for several classes of graphs. Next we show that for every non-negative integer there exists a tree with such 2-bondage number. Finally, we characterize all trees with 2-bondage number equaling one or two.

Keywords:

• 2-domination
• bondage
• 2-bondage
• tree

2010 AMS Subject Classifications:

• 05C05
• 05C69

Acknowledgements

The research was supported by the Polish National Science Centre grant 2011/02/A/ST6/00201.