On Bondage Numbers of Graphs: A Survey with Some Comments

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.

Minimum edge blocker dominating set problem

“…The minimum vertex/edge blocker and the most vital vertices/ edges problems have been studied in literature with respect to different graph properties, such as connectivity (Addis, Di Summa, & Grosso, 2013;Arulselvan, Commander, Elefteriadou, & Pardalos, 2009;Di Summa, Grosso, & Locatelli, 2011;Shen, Smith, & Goli, 2012;Veremyev, Prokopyev, & Pasiliao, 2014), shortest path (Bar-Noy et al, 1995;Israeli & Wood, 2002;Khachiyan et al 2008), maximum flow (Altner, Ergun, & Uhan, 2010;Ghare, Montgomery, & Turner, 1971;Wollmer, 1964;Wood, 1993), spanning tree (Bazgan, Toubaline, & Vanderpooten, 2012;2013;Frederickson & Solis-Oba, 1996), assignment (Bazgan, Toubaline, & Vanderpooten, 2010b), 1-median (Bazgan et al, 2010a), 1-center (Bazgan et al, 2010a), matching (Ries et al, 2010;Zenklusen, 2010;Zenklusen et al, 2009), independent sets (Bazgan et al, 2011), vertex covers (Bazgan et al, 2011), andcliques (Mahdavi Pajouh, Boginski, &.…”

“…Hu and Xu (2012) showed that the problem of determining the bondage number of a graph is NP-hard. Upper bounds and lower bounds on the bondage number of a graph along with different generalizations of this concept have also been proposed in the literature (Xu, 2013).…”

The Roman bondage number of a digraph

“…Recently, Carlson and Develin [3], Shan and Kang [16], and Huang and Xu [10,12] studied the bondage number for digraphs, independently. A thorough study of bondage number appears in [19].…”

Bondage number of grid graphs

“…In [8], the exact values of bondage number of Cartesian product of two paths P n and P m have been determined for m ≤ 4. For more results on bondage number of a graph we suggest the survey [15].…”