Abstract
The domination number of a graph is the smallest number of vertices which dominate all remaining vertices by edges of . The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of . The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations. This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers.
Highlights
For terminology and notation on graph theory not given here, the reader is referred to Xu [1]
Since the bondage number is defined as the smallest number of edges whose removal results in an increase of domination number, each constructive method that creates a concrete bondage set leads to an upper bound on the bondage number
It is quite difficult to determine the exact value of the bondage number for a given graph since it strongly depends on the dominating number of the graph
Summary
For terminology and notation on graph theory not given here, the reader is referred to Xu [1]. A measure of the efficiency of a domination in graphs was first given by Bauer et al [7] in 1983, who called this measure as domination line-stability, defined as the minimum number of lines (i.e., edges) which when removed from G increases γ. The bondage number b(G) of a nonempty undirected graph G is the minimum number of edges whose removal from G results in a graph with larger domination number. If B is a minimum bondage set, γ(G − B) = γ(G) + 1 because the removal of one single edge cannot increase the domination number by more than one. We introduce some results for vertex-transitive graphs by applying efficient dominating sets
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