scholarly journals An improved upper bound for the bondage number of graphs on surfaces

2012 ◽  
Vol 312 (18) ◽  
pp. 2776-2781 ◽  
Author(s):  
Jia Huang
Keyword(s):  
Upper Bound ◽  
Bondage Number ◽  
Graphs On Surfaces

The k-power bondage number of a graph

2016 ◽  
Vol 08 (04) ◽  
pp. 1650064
Author(s):  
Seethu Varghese ◽  
A. Vijayakumar
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Power Domination ◽  
Bondage Number ◽  
Number Formula

The [Formula: see text]-power domination number, [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of a [Formula: see text]-power dominating set of [Formula: see text]. In this paper, we initiate the study of the [Formula: see text]-power bondage number, [Formula: see text], of a graph [Formula: see text], i.e., the minimum cardinality among all sets [Formula: see text] for which [Formula: see text]. We obtain a sharp upper bound for [Formula: see text] in terms of the degree of [Formula: see text]. We prove that [Formula: see text] for any nonempty tree [Formula: see text] and also provide some conditions on [Formula: see text] for [Formula: see text].

scholarly journals The Bondage Number of Random Graphs

10.37236/5180 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Side Product ◽  
Bondage Number ◽  
Concentration Result

A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The bondage number of a nonempty graph $G$ is the size of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. In this note, we study the bondage number of the binomial random graph $G(n,p)$. We obtain a lower bound that matches the order of the trivial upper bound. As a side product, we give a one-point concentration result for the domination number of $G(n,p)$ under certain restrictions.

Sign in / Sign up

Export Citation Format

Share Document