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International Journal of Combinatorics
Volume 2013 (2013), Article ID 595210, 34 pages
http://dx.doi.org/10.1155/2013/595210
Review Article

On Bondage Numbers of Graphs: A Survey with Some Comments

School of Mathematical Sciences, University of Science and Technology of China, Wentsun Wu Key Laboratory of CAS, Hefei, Anhui 230026, China

Received 15 December 2012; Revised 20 February 2013; Accepted 11 March 2013

Academic Editor: Chính T. Hoang

Copyright © 2013 Jun-Ming Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

For terminology and notation on graph theory not given here, the reader is referred to Xu [1]. Let be a finite, undirected, and simple graph. We call and the order and size of and denote them by and , respectively, unless otherwise specified. Through this paper, the notations , , and always denote a path, a cycle, and a complete graph of order , respectively, the notation denotes a complete -partite graph with and , with , and is a star. For two vertices and in a connected graph , we use to denote the distance between and in .

For a vertex in , let be the open set of neighbors of and the closed set of neighbors of . For a subset , , and , where . Let be the set of edges incident with in ; that is, . We denote the degree of by . The maximum and the minimum degrees of are denoted by and , respectively. A vertex of degree zero is called an isolated vertex. An edge incident with a vertex of degree one is called a pendant edge.

The bondage number is an important parameter of graphs which is based upon the well-known domination number.

A subset is called a dominating set of if ; that is, every vertex in has at least one neighbor in . The domination number of , denoted by , is the minimum cardinality among all dominating sets; that is, A dominating set is called a -set of if .

The domination is such an important and classic conception that it has become one of the most widely studied topics in graph theory and also is frequently used to study property of networks. The domination, with many variations and generalizations, is now well studied in graph and networks theory. The early vast literature on domination includes the bibliography compiled by Hedetniemi and Laskar [2] and a thorough study of domination appears in the books by Haynes et al. [3, 4]. However, the problem determining the domination number for general graphs was early proved to be NP-complete (see GT2 in Appendix in Garey and Johnson [5], 1979).

Among various problems related with the domination number, some focus on graph alterations and their effects on the domination number. Here, we are concerned with a particular graph alternation, the removal of edges from a graph.

Graphs with domination numbers changed upon the removal of an edge were first investigated by Walikar and Acharya [6] in 1979. A graph is called edge-domination-critical graph if for every edge in . The edge-domination-critical graph was characterized by Bauer et al. [7] in 1983; that is, a graph is edge-domination-critical if and only if it is the union of stars.

However, for lots of graphs, the domination number is out of the range of one-edge removal. It is immediate that for any spanning subgraph of . Every graph has a spanning forest with , and so, in general, a graph has a nonempty subset for which .

Then it is natural for the alternation to be generalized to the removal of several edges, which is just enough to enlarge the domination number. That is the idea of the bondage number.

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 increases .

In 1990, Fink et al. [8] formally introduced the bondage number as a parameter for measuring the vulnerability of the interconnection network under link failure. The minimum dominating set of sites plays an important role in the network for it dominates the whole network with the minimum cost. So we must consider whether its function remains good when the network is attacked. Suppose that someone such as a saboteur does not know which sites in the network take part in the dominating role, but does know that the set of these special sites corresponds to a minimum dominating set in the related graph. Then how many links does he has to attack so that the cost cannot remains the same in order to dominate the whole network? That minimum number of links is just the bondage number.

The bondage number of a nonempty undirected graph is the minimum number of edges whose removal from results in a graph with larger domination number. The precise definition of the bondage number is as follows: Since the domination number of every spanning subgraph of a nonempty graph is at least as great as , the bondage number of a nonempty graph is well defined.

We call such an edge-set with the bondage set and the minimum one the minimum bondage set. In fact, if is a minimum bondage set, then because the removal of one single edge cannot increase the domination number by more than one. If does not exist, for example, empty graphs, we define .

It is quite difficult to compute the exact value of the bondage number for general graphs since it strongly depends on the domination number of the graphs. Much work focused on the bounds on the bondage number as well as the restraints on some particular classes of graphs or networks. The purpose of this paper is to give a survey of results and research methods related to these topics for graphs and digraphs. For some results and research methods, we will make some comments to develop our further study.

The rest of the paper is organized as follows. Section 2 gives some preliminary results and complexity. Sections 3 and 4 survey some upper bounds and lower bounds, respectively. The results for some special classes of graphs and planar graphs are stated in Sections 5 and 6, respectively. In Section 7, we introduce some results on crossing number restraints. In Sections 8 and 9, we are concerned about other and generalized types of bondage numbers, respectively. In Section 10, we introduce some results for digraphs. In the last section, we introduce some results for vertex-transitive graphs by applying efficient dominating sets.

2. Simplicity and Complexity

As we have known from Introduction, the bondage number is an important parameter for measuring the stability or the vulnerability of a domination in a graph or a network. Our aim is to compute the bondage number for any given graphs or networks. One has determined the exact value of the bondage number for some graphs with simple structure. For arbitrarily given graph, however, it has been proved that determining its bondage number is NP-hard.

2.1. Exact Values for Ordinary Graphs

We begin our investigation of the bondage number by computing its value for several well-known classes of graphs with simple structure. In 1990, Fink et al. [8] proposed the concept of the bondage number and completely determined the exact values of bondage numbers of some ordinary graphs, such as complete graphs, paths, cycles, and complete multipartite graphs.

Theorem 1 (Fink et al. [8], 1990). The exact values of bondage numbers of the following class of graphs are completely determined:

The complete graph is the unique -regular graph of order by Theorem 1. The -partite graph with is an -regular graph of order , and not unique, by Theorem 1 for an even integer . For an -regular graph of order , Hu and Xu [9] obtained the following result.

Theorem 2 (Hu and Xu [9]). for any ()-regular graph of order .

Up to the present, no results have been known for -regular graphs with . For general graphs, there are the following two results.

Theorem 3 (Teschner [10], 1997). If is a nonempty graph with a unique minimum dominating set, then .

Theorem 4 (Bauer et al. [7], 1983). If any vertex of a graph is adjacent with two or more vertices of degree one, then .

Bauer et al. [7] observed that the star is the unique graph with the property that the bondage number is 1 and the deletion of any edge results in the domination number increasing. Motivated by this fact, Hartnell and Rall [11] proposed the concept of uniformly bonded graphs. A graph is called to be uniformly bonded if it has bondage number and the deletion of any edges results in a graph with increased domination number. Unfortunately, there are a few uniformly bonded graphs.

Theorem 5 (Hartnell and Rall [11], 1999). The only uniformly bonded graphs with bondage number 2 are and . The unique uniformly bonded graph with bondage number 3 is . There are no such graphs for bondage number greater than 3.

As we mentioned, to compute the exact value of bondage number for a graph strongly depends upon its domination number. In this sense, studying the bondage number can greatly inspire one's research concerned with dominations. However, determining the exact value of domination number for a given graph is quite difficult. In fact, even if the exact value of the domination number for some graph is determined, it is still very difficulty to compute the value of the bondage number for that graph. For example, for the hypercube , we have , but we have not yet determined for any .

Perhaps Theorems 3 and 4 provide an approach to compute the exact value of bondage number for some graphs by establishing some sufficient conditions for . In fact, we will see later that Theorem 3 plays an important role in determining the exact values of the bondage numbers for some graphs. Thus, to study the bondage number, it is important to present various characterizations of graphs with a unique minimum dominating set.

2.2. Characterizations of Trees

For trees, Bauer et al. [7] in 1983 from the point of view of the domination line-stability, independently and Fink et al. [8] in 1990 from the point of view of the domination edge-vulnerability obtained the following result.

Theorem 6. For any nontrivial tree , .

By Theorem 6, it is natural to classify all trees according to their bondage numbers. Fink et al. [8] proved that a forbidden subgraph characterization to classify trees with different bondage numbers is impossible, since they proved that if is a forest, then is an induced subgraph of a tree with and a tree with . However, they pointed out that the complexity of calculating the bondage number of a tree of order is at most by methodically removing each pair of edges.

Even so, some characterizations, whether a tree has bondage number 1 or 2, have been found by several authors; see, for example, [10, 12, 13].

First we describe the method due to Hartnell and Rall [12], by which all trees with bondage number 2 can be constructed inductively. An important tree in the construction is shown in Figure 1. To characterize this construction, we need some terminologies:

595210.fig.001
Figure 1: Tree .
(1)attach a path to a vertex of a tree, which means to link and one end-vertex of the by an edge.(2)attach to a vertex , which means to link and a vertex of by an edge.

The following are four operations on a tree :Type 1: attach a to , where and belongs to at least one -set of (such a vertex exists, say, one end-vertex of ).Type 2: attach a to , where .Type 3: attach to , where belongs to at least one -set of .Type 4: attach , , to , where can be any vertex of .

Let is a tree, and , and for some , or can be obtained from or by a finite sequence of operations of Types .

Theorem 7 (Hartnell and Rall [12], 1992). A tree has bondage number 2 if and only if it belongs to .

Looking at different minimum dominating sets of a tree, Teschner [10] presented a totally different characterization of the set of trees having bondage number 1. They defined a vertex to be universal if it belongs to each minimum dominating set, and to be idle if it does not belong to any minimum dominating set.

Theorem 8 (Teschner [10], 1997). A tree has bondage number 1 if and only if has a universal vertex or an edge satisfying(1) and are neither universal nor idle; and(2)all neighbors of and (except for and ) are idle.

For a positive integer , a subset is called a -independent set (also called a -packing) if for any two distinct vertices and in . When , -set is the normal independent set. The maximum cardinality among all -independent sets is called the -independence number (or -packing number) of , denoted by . A -independent set is called an -set if . A graph is said to be -stable if for every edge of . There are two important results on -independent sets.

Proposition 9 (Topp and Vestergaard [13], 2000). A tree is -stable if and only if has a unique -set.

Proposition 10 (Meir and Moon [14], 1975). for any connected graph with equality for any tree.

Hartnell et al. [15], independently and Topp and Vestergaard [13] also gave a constructive characterization of trees with bondage number 2 and, applying Proposition 10, presented another characterization of those trees.

Theorem 11 (Hartnell et al. [15], 1998; Top and Vestergaard [13], 2000). for a tree if and only if has a unique -set.

According to this characterization, Hartnell et al. [15] presented a linear algorithm for determining the bondage number of a tree.

In this subsection, we introduce three characterizations for trees with bondage number 1 or 2. The characterization in Theorem 7 is constructive, constructing all trees with bondage number 2, a natural and straightforward method, by a series of graph-operations. The characterization in Theorem 8 is a little advisable, by describing the inherent property of trees with bondage number 1. The characterization in Theorem 11 is wonderful, by using a strong graph-theoretic concept, -set. In fact, this characterization is a byproduct of some results related to -sets for trees. It is that this characterization closed the relation between two concepts, the bondage number and the -independent set, and hence is of research value and important significance.

2.3. Complexity for General Graphs

As mentioned above, the bondage number of a tree can be determined by a linear time algorithm. According to this algorithm, we can determine within polynomial time the domination number of any tree by removing each edge and verifying whether the domination number is enlarged according to the known linear time algorithm for domination numbers of trees.

However, it is impossible to find a polynomial time algorithm for bondage numbers of general graphs. If such an algorithm exists, then the domination number of any nonempty undirected graph can be determined within polynomial time by repeatedly using . Let and , where is the minimum edge set of found by such that for each ; we can always find the minimum whose removal from enlarges the domination number, until is empty for some , though is not empty. Then . As known to all, if , the minimum dominating set problem is NP-complete, and so polynomial time algorithms for the bondage number do not exist unless .

In fact, Hu and Xu [16] have recently shown that the problem determining the bondage number of general graphs is NP-hard.

Problem 1. Consider the decision problem:Bondage ProblemInstance: a graph and a positive integer .Question: is ?

Theorem 12 (Hu and Xu [16], 2012). The bondage problem is NP-hard.

The basic way of the proof is to follow Garey and Johnson's techniques for proving NP-hardness [5] by describing a polynomial transformation from the known NP-complete problem: -satisfiability problem.

Theorem 12 shows that we are unable to find a polynomial time algorithm to determine bondage numbers of general graphs unless . At the same time, this result also shows that the following study on the bondage number is of important significance.(i)Find approximation polynomial algorithms with performance ratio as small as possible.(ii)Find the lower and upper bounds with difference as small as possible.(iii)Determine exact values for some graphs, specially well-known networks.

Unfortunately, we cannot prove whether or not determining the bondage number is NP-problem since for any subset it is not clear that there is a polynomial algorithm to verify . Since the problem of determining the domination number is NP-complete, we conjecture that it is not in NP. This is a worthwhile task to be studied further.

Motivated by the linear time algorithm of Hartnell et al. to compute the bondage number of a tree, we can make an attempt to consider whether there is a polynomial time algorithm to compute the bondage number for some special classes of graphs such as planar graphs, Cayley graphs, or graphs with some restrictions of graph-theoretical parameters such as degree, diameter, connectivity, and domination number.

3. Upper Bounds

By Theorem 12, since we cannot find a polynomial time algorithm for determining the exact values of bondage numbers of general graphs, it is weightily significative to establish some sharp bounds on the bondage number of a graph. In this section, we survey several known upper bounds on the bondage number in terms of some other graph-theoretical parameters.

3.1. Most Basic Upper Bounds

We start this subsection with a simple observation.

Observation 1 (Teschner [10], 1997). Let be a spanning subgraph obtained from a graph by removing edges. Then .

If we select a spanning subgraph such that , then Observation 1 yields some upper bounds on the bondage number of a graph . For example, take and , where , the following result can be obtained.

Theorem 13 (Bauer et al. [7], 1983). If there exists at least one vertex in a graph such that , then .

The following early result obtained can be derived from Observation 1 by taking and .

Theorem 14 (Bauer et al. [7], 1983; Fink et al. [8], 1990). for any two adjacent vertices and in a graph ; that is,

This theorem gives a natural corollary obtained by several authors.

Corollary 15 (Bauer et al. [7], 1983; Fink et al. [8], 1990). If is a graph without isolated vertices, then .

In 1999, Hartnell and Rall [11] extended Theorem 14 to the following more general case, which can be also derived from Observation 1 by taking if , where is a path of length 2 in .

Theorem 16 (Hartnell and Rall [11], 1999). for any distinct two vertices and in a graph with ; that is,

Corollary 17 (Fink et al. [8], 1990). If a vertex of a graph is adjacent with two or more vertices of degree one, then .

We remark that the bounds stated in Corollary 15 and Theorem 16 are sharp. As indicated by Theorem 1, one class of graphs in which the bondage number achieves these bounds is the class of cycles whose orders are congruent to 1 modulo 3.

On the other hand, Hartnell and Rall [17] sharpened the upper bound in Theorem 14 as follows, which can be also derived from Observation 1.

Theorem 18 (Hartnell and Rall [17], 1994). for any two adjacent vertices and in a graph ; that is,

These results give simple but important upper bounds on the bondage number of a graph, and is also the foundation of almost all results on bondage numbers upper bounds obtained till now.

By careful consideration of the nature of the edges from the neighbors of and , Wang [18] further refined the bound in Theorem 18. For any edge , contains the following four subsets:(1);(2);(3) for some ;(4).

The illustrations of , , , and are shown in Figure 2 (corresponding vertices pointed by dashed arrows).

595210.fig.002
Figure 2: Illustration of for .

Theorem 19 (Wang [18], 1996). For any nonempty graph ,

The graph in Figure 2 shows that the upper bound given in Theorem 19 is better than that in Theorems 16 and 18, for the upper bounds obtained from these two theorems are and , respectively, while the upper bound given by Theorem 19 is .

The following result is also an improvement of Theorem 14, in which .

Theorem 20 (Teschner [10], 1997). If contains a complete subgraph with , then .

Following Fricke et al. [19], a vertex of a graph is -good if belongs to some -set of and -bad if belongs to no -set of . Let be the set of -good vertices, and let be the set of -bad vertices in . Clearly, is a partition of . Note there exists some such that , say, one end-vertex of . Samodivkin [20] presented some sharp upper bounds for in terms of -good and -bad vertices of .

Theorem 21 (Samodivkin [20], 2008). Let be a graph.(a)Let . If , then (b)If , then

Theorem 22 (Samodivkin [20], 2008). Let be a graph. If and , then

Proposition 23 (Samodivkin [20], 2008). Under the notation of Theorem 19, if , then .

By Proposition 23, if , then Hence, Theorem 19 can be seen to follow from Theorem 22. Any graph with achieving the upper bound in Theorem 19 can be used to show that the bound in Theorem 22 is sharp.

Let be an integer. Samodivkin [20] constructed a very interesting graph to show that the upper bound in Theorem 22 is better than the known bounds. Let be mutually vertex-disjoint graphs such that , and for each . Let , and for each . The graph is defined as follows: Such a constructed graph is shown in Figure 3 when .

595210.fig.003
Figure 3: The graph .

Observe that , , , , , and for each . Moreover, and for any . Hence, each of the bounds stated in Theorems 1320 is greater than or equals .

Consider the graph . Clearly, and Therefore, which implies that the upper bound stated in Theorem 22 is equal to . Clearly, , and hence this bound is sharp for .

From the graph , we obtain the following statement immediately.

Proposition 24. For every integer , there is a graph such that the difference between any upper bound stated in Theorems 1320 and the upper bound in Theorem 22 is equal to .

Although Theorem 22 supplies us with the upper bound that is closer to for some graph than what any one of Theorems 1320 provides, it is not easy to determine the sets and mentioned in Theorem 22 for an arbitrary graph . Thus, the upper bound given in Theorem 22 is of theoretical importance, but not applied since, until now, we have not found a new class of graphs whose bondage numbers are determined by Theorem 22.

The above-mentioned upper bounds on the bondage number are involved in only degrees of two vertices. Hartnell and Rall [11] established an upper bound on in terms of the numbers of vertices and edges of with order . For any connected graph , let represent the average degree of vertices in ; that is, . Hartnell and Rall first discovered the following proposition.

Proposition 25. For any connected graph , there exist two vertices and with distance at most two and with the property that .

Using Proposition 25 and Theorem 16, Hartnell and Rall gave the following bound.

Theorem 26 (Hartnell and Rall [11], 1999). for any connected graph .

Note that for any graph with order and size . Theorem 26 implies the following bound in terms of order and size .

Corollary 27. for any connected graph with order and size .

A lower bound on in terms of order and the bondage number is obtained from Corollary 28 immediately.

Corollary 28. for any connected graph with order and size .

Hartnell and Rall [11] gave some graphs with order to show that for each value of ; the lower bound on given in the Corollary 28 is sharp for some values of .

3.2. Bounds Implied by Connectivity

Use and to denote the vertex-connectivity and the edge-connectivity of a connected graph , respectively, which are the minimum numbers of vertices and edges whose removal result in disconnected. The famous Whitney inequality is stated as for any graph or digraph . Corollary 15 is improved by several authors as follows.

Theorem 29 (Hartnell and Rall [17], 1994; Teschner [10], 1997). If is a connected graph, then .

The upper bound given in Theorem 29 can be attained. For example, a cycle with , by Theorem 1. Since , we have .

Motivated by Corollary 15, Theorems 29, and the Whitney inequality, Dunbar et al. [21] naturally proposed the following conjecture.

Conjecture 30. If is a connected graph, then .

However, Liu and Sun [22] presented a counterexample to this conjecture. They first constructed a graph showed in Figure 4 with and . Then, let be the disjoint union of two copies of by identifying two vertices of degree two. They proved that . Clearly, is a -regular graph with and , and so by Theorem 29. Thus, .

595210.fig.004
Figure 4: A graph with and .

With a suspicion of the relationship between the bondage number and the vertex-connectivity of a graph, the following conjecture is proposed.

Conjecture 31 (Liu and Sun [22], 2003). For any positive integer , there exists a connected graph such that .

To the knowledge of the author, until now no results have been known about this conjecture.

We conclude this subsection with the following remarks. From Theorem 29, if Conjecture 31 holds for some connected graph , then , which implies that is of large edge-connectivity and small vertex-connectivity. Use to denote the minimum edge-degree of ; that is, Theorem 14 implies the following result.

Proposition 32. for any graph .

Use to denote the restricted edge-connectivity of a connected graph , which is the minimum number of edges whose removal result in disconnected and no isolated vertices.

Proposition 33 (Esfahanian and Hakimi [23], 1988). If is neither nor , then .

Combining Propositions 32 and 33, we propose a conjecture.

Conjecture 34. If a connected is neither nor , then .

A cycle also satisfies since for any integer .

For the graph shown in Figure 4, and , and so .

For the -regular graph constructed by Liu and Sun [22] obtained from the disjoint union of two copies of the graph showed in Figure 4 by identifying two vertices of degree two, we have . Clearly, . Thus, .

For the -regular graph constructed by Samodivkin [20], see Figure 3 for , . Clearly, and . Thus, .

These examples show that if Conjecture 34 is true then the given upper bound is tight.

3.3. Bounds Implied by Degree Sequence

Now let us return to Theorem 16, from which Teschner [10] obtained some other bounds in terms of the degree sequences. The degree sequence of a graph with vertex-set is the sequence with , where for each . The following result is essentially a corollary of Theorem 16.

Theorem 35 (Teschner [10], 1997). Let be a nonempty graph with degree sequence . If , then .

Combining Theorem 35 with Proposition 10 (i.e., ), we have the following corollary.

Corollary 36 (Teschner [10], 1997). Let be a nonempty graph with the degree sequence . If , then .

In [10], Teschner showed that these two bounds are sharp for arbitrarily many graphs. Let , where is a cycle for any integer . Then and so by Corollary 36. Since is a spanning subgraph of and , Observation 1 yields that , and so .

Although various upper bounds have been established as the above, we find that the appearance of these bounds is essentially based upon the local structures of a graph, precisely speaking, the structures of the neighborhoods of two vertices within distance 2. Even if these bounds can be achieved by some special graphs, it is more often not the case. The reason lies essentially in the definition of the bondage number, which is the minimum value among all bondage sets, an integral property of a graph. While it easy to find upper bounds just by choosing some bondage set, the gap between the exact value of the bondage number and such a bound obtained only from local structures of a graph is often large. For example, a star , however large is, . Therefore, one has been longing for better bounds upon some integral parameters. However, as what we will see below, it is difficult to establish such upper bounds.

3.4. Bounds in -Critical Graphs

A graph is called a vertex domination-critical graph (vc-graph or -critical for short) if for any , proposed by Brigham et al. [24] in 1988. Several families of graphs are known to be -critical. From definition, it is clear that if is a -critical graph, then . The class of -critical graphs with is characterized as follows.

Proposition 37 (Brigham et al. [24], 1988). A graph with is a -critical graph if and only if is a complete graph () with a perfect matching removed.

A more interesting family is composed of the -critical graphs defined for by the circulant undirected graph , where and , in which and .

The reason why -critical graphs are of special interest in this context is easy to see that they play an important role in study of the bondage number. For instance, it immediately follows from Theorem 13 that if then is a -critical graph. The -critical graphs are defined exactly in this way. In order to find graphs with a high bondage number (i.e., higher than and beyond its general upper bounds for the bondage number, we, therefore, have to look at -critical graphs.

The bondage numbers of some -critical graphs have been examined by several authors; see, for example, [10, 20, 25, 26]. From Theorem 13, we know that the bondage number of a graph is bounded from above by if is not a -critical graph. For -critical graphs, it is more difficult to find an upper bound. We will see that the bondage numbers of -critical graphs in general are not even bounded from above by for any fixed natural number .

In this subsection, we introduce some upper bounds for the bondage number of a -critical graph. By Proposition 37, we easily obtain the following result.

Theorem 38. If is a -critical graph with , then .

In Section 4, by Theorem 59, we will see that the equality in Theorem 38 holds; that is, if is a -critical graph with .

Theorem 39 (Teschner [10], 1997). Let be a -critical graph with degree sequence . Then , where .

As we mentioned above, if is a -critical graph with , then , which shows that the bound given in Theorem 39 can be attained for . However, we have not known whether this bound is tight for general . Theorem 39 gives the following corollary immediately.

Corollary 40 (Teschner [10], 1997). Let be a -critical graph with degree sequence . If , then .

From Theorem 38 and Corollary 40, we have if is a -critical graph with . The following result shows that this bound is not tight.

Theorem 41 (Teschner [26], 1995). Let be a -critical graph graph with . Then .

Until now, we have not known whether the bound given in Theorem 41 is tight or not. We state two conjectures on -critical graphs proposed by Samodivkin [20]. The first of them is motivated by Theorems 22 and 19.

Conjecture 42 (Samodivkin [20], 2008). For every connected nontrivial -critical graph ,

To state the second conjecture, we need the following result on -critical graphs.

Proposition 43. If is a -critical graph, then ; moreover, if the equality holds, then is regular.

The upper bound in Proposition 43 is sharp in the sense that equality holds for the infinite class of -critical graphs defined in the beginning of this subsection. In Proposition 43, the first result is due to Brigham et al. [24] in 1988; the second is due to Fulman et al. [27] in 1995.

For a -critical graph with , by Proposition 43, .

Theorem 44 (Teschner [26], 1995). If is a -critical graph with and , then .

We have not known yet whether the equality in Theorem 44 holds or not. However, Samodivkin proposed the following conjecture.

Conjecture 45 (Samodivkin [20], 2008). If is a -critical graph with vertices, then .

In general, based on Theorem 41, Teschner proposed the following conjecture.

Conjecture 46 (Teschner [26], 1995). If is a -critical graph, then for .

We conclude this subsection with some remarks. Graphs which are minimal or critical with respect to a given property or parameter frequently play an important role in the investigation of that property or parameter. Not only are such graphs of considerable interest in their own right, but also a knowledge of their structure often aids in the development of the general theory. In particular, when investigating any finite structure, a great number of results are proven by induction. Consequently, it is desirable to learn as much as possible about those graphs that are critical with respect to a given property or parameter so as to aid and abet such investigations.

In this subsection, we survey some results on the bondage number for -critical graphs. Although these results are not very perfect, they provides a feasible method to approach the bondage number from different viewpoints. In particular, the methods given in Teschner [26] worth further exploration and development.

The following proposition is maybe useful for us to further investigate the bondage number of a -critical graph.

Proposition 47 (Brigham et al. [24], 1988). If has a nonisolated vertex such that the subgraph induced by is a complete graph, then is not -critical.

This simple fact shows that any -critical graph contains no vertices of degree one.

3.5. Bounds Implied by Domination

In the preceding subsection, we introduced some upper bounds on the bondage numbers for -critical graphs by consideration of dominations. In this subsection, we introduce related results for a general graph with given domination number.

Theorem 48 (Fink et al. [8], 1990). For any connected graph of order ,(a);(b) if ;(c).

While the upper bound of on is not particularly good for many classes of graphs (e.g., trees and cycles), it is an attainable bound. For example, if is a complete -partite graph for and an even integer, then the three bounds on in Theorem 48 are sharp.

Teschner [26] consider a graph with or . The next result is almost trivial but useful.

Lemma 49. Let be a graph with order and , the number of vertices of degree . Then .

Since clearly, Lemma 49 yields the following result immediately.

Theorem 50 (Teschner [26], 1995). for any graph with .

For a complete graph , , which shows that the upper bound given in Theorem 50 can be attained. For a graph with , by Theorem 59 later, the upper bound given in Theorem 48 (b) can be also attained by a -critical graph (see Proposition 37).

3.6. Two Conjectures

In 1990, when Fink et al. [8] introduced the concept of the bondage number; they proposed the following conjecture.

Conjecture 51. for any nonempty graph .

Although some results partially support Conjecture 51, Teschner [28] in 1993 found a counterexample to this conjecture, the cartesian product , which shows that .

If a graph is a counterexample to Conjecture 51, it must be a -critical graph by Theorem 13. That is why the vertex domination-critical graphs are of special interest in the literature.

Now, we return to Conjecture 51. Hartnell and Rall [17] and Teschner [29], independently, proved that can be much greater than by showing the following result.

Theorem 52 (Hartnell and Rall [17], 1994; Teschner [29]. 1996). For an integer , let be the cartesian product . Then .

This theorem shows that there exist no upper bounds of the form for any integer . Teschner [26] proved that for any graph with (see Theorems 50 and 48) and for -critical graphs with and proposed the following conjecture.

Conjecture 53 (Teschner [26], 1995). for any graph .

We believe that this conjecture is true, but so far no much work about it has been known.

4. Lower Bounds

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. For that reason, it is hard to find lower bounds. Nevertheless, there are still a few lower bounds.

4.1. Bounds Implied by Subgraphs

The first lower bound on the bondage number is gotten in terms of its spanning subgraph.

Theorem 54 (Teschner [10], 1997). Let be a spanning subgraph of a nonempty graph . If , then .

By Theorem 1, if and , otherwise if and otherwise. From these results and Theorem 54, we get the following two corollaries.

Corollary 55. If is hamiltonian with order and , then and in addition if .

Corollary 56. If with order has a hamiltonian path and , then if .

4.2. Other Bounds

The vertex covering number of is the minimum number of vertices that are incident with all edges in . If has no isolated vertices, then clearly. In [30], Volkmann gave a lot of graphs with .

Theorem 57 (Teschner [10], 1997). Let be a graph. If , then(a);(b) if is a -critical graph.

The graph shown in Figure 5 shows that the bound given in Theorem 57(b) is sharp. For the graph , it is a -critical graph and . By Theorem 57, we have . On the other hand, by Theorem 16. Thus, .

595210.fig.005
Figure 5: A -critical graph with , and .

Proposition 58 (Sanchis [31], 1991). Let be a graph of order and size . If has no isolated vertices and , then .

Using the idea in the proof of Theorem 57, every upper bound for can lead to a lower bound for . In this way, Teschner [10] obtained another lower bound from Proposition 58.

Theorem 59 (Teschner [10], 1997). Let be a graph of order and size . If , then(a);(b) if is a -critical graph.

The lower bound in Theorem 59(b) is sharp for a class of -critical graphs with domination number 2. Let be the graph obtained from complete graph () by removing a perfect matching. By Proposition 37, is a -critical graph with . Then by Theorem 59 and by Theorem 38, and so .

As far as the author knows, there are no more lower bounds in the present literature. In view of applications of the bondage number, a network is vulnerable if its bondage number is small while it is stable if its bondage number is large. Therefore, better lower bounds let us learn better about the stability of networks from this point of view. Thus, it is of great significance to seek more lower bounds for various classes of graphs.

5. Results on Graph Operations

Generally speaking, 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. Thus, determining bondage numbers for some special graphs is interesting if the dominating numbers of those graphs are known or can be easily determined. In this section, we will introduce some known results on bondage numbers for some special classes of graphs obtained by some graph operations.

5.1. Cartesian Product Graphs

Let and be two graphs. The Cartesian product of and is an undirected graph, denoted by , where ; two distinct vertices and , where and , are linked by an edge in if and only if either and , or and . The Cartesian product is a very effective method for constructing a larger graph from several specified small graphs.

Theorem 60 (Dunbar et al. [21], 1998). For ,

For the Cartesian product of two cycles and , where and , Klavžar and Seifter [32] determined for some and . For example, for and for . Kim [33] determined and . For a general , the exact values of the bondage numbers of and are determined as follows.

Theorem 61 (Sohn et al. [34], 2007). For ,

Theorem 62 (Kang et al. [35], 2005). for .

For larger and , Huang and Xu [36] obtained the following result; see Theorem 197 for more details.

Theorem 63 (Huang and Xu [36], 2008). for any positive integers and .

Xiang et al. [37] determined that for ,

For the Cartesian product of two paths and , Jacobson and Kinch [38] determined , , and if , and otherwise. The bondage number of for and is determined as follows (Li [39] also determined ).

Theorem 64 (Hu and Xu [40], 2012). For ,

From the proof of Theorem 64, we find that if and , then the removal of the vertex in does not change the domination number. If increases, the effect of for the domination number will be smaller and smaller from the probability. Therefore, we expect it is possible that for and give the following conjecture.

Conjecture 65. for and .

5.2. Block Graphs and Cactus Graphs

In this subsection, we introduce some results for corona graphs, block graphs, and cactus graphs.

The corona , proposed by Frucht and Harary [41], is the graph formed from a copy of and copies of by joining the th vertex of to the th copy of . In particular, we are concerned with the corona , the graph formed by adding a new vertex , and a new edge for every vertex in .

Theorem 66 (Carlson and Develin [42], 2006). .

This is a very important result, which is often used to construct a graph with required bondage number. In other words, this result implies that for any given positive integer there is a graph such that .

A block graph is a graph whose blocks are complete graphs. Each block in a cactus graph is either a cycle or a . If each block of a graph is either a complete graph or a cycle, then we call this graph a block-cactus graph.

Theorem 67 (Teschner [43], 1997). for any block graph .

Teschner [43] characterized all block graphs with . In the same paper, Teschner found that -critical graphs are instrumental in determining bounds for the bondage number of cactus and block graphs and obtained the following result.

Theorem 68 (Teschner [43], 1997). for any nontrivial cactus graph .

This bound can be achieved by where is a nontrivial cactus graph without vertices of degree one by Theorem 66. In 1998, Dunbar et al. [21] proposed the following problem.

Problem 2. Characterize all cactus graphs with bondage number 3.

Some upper bounds for block-cactus graphs were also obtained.

Theorem 69 (Dunbar et al. [21], 1998). Let be a connected block-cactus graph with at least two blocks. Then .

Theorem 70 (Dunbar et al. [21], 1998). Let be a connected block-cactus graph which is neither a cactus graph nor a block graph. Then .

5.3. Edge-Deletion and Edge-Contraction

In order to obtain further results of the bondage number, we may consider how the bondage number changes under some operations of a graph , which remain the domination number unchanged or preserve some property of , such as planarity. Two simple operations satisfying these requirements are the edge-deletion and the edge-contraction.

Theorem 71 (Huang and Xu [44], 2012). Let be any graph and . Then . Moreover, if .

From Theorem 1, for any in , which shows that the lower bound on is sharp. The upper bound can reach for any graph with .

Next, we consider the effect of the edge-contraction on the bondage number. Given a graph , the contraction of by the edge , denoted by , is the graph obtained from by contracting two vertices and to a new vertex and then deleting all multiedges. It is easy to observe that for any graph , for any edge of .

Theorem 72 (Huang and Xu [44], 2012). Let be any graph and be any edge in . If and , then .

We can use examples and to show that the conditions of Theorem 72 are necessary and the lower bound in Theorem 72 is tight. Clearly, and ; for any edge , and . By Theorem 1, if (mod ), then and . Thus, the result in Theorem 72 is generally invalid without the hypothesis . Furthermore, the condition cannot be omitted even if , since for odd , , by Theorem 1.

On the other hand, if (mod ) and if is even, which shows that the equality in may hold. Thus, the bound in Theorem 72 is tight. However, provided all the conditions, can be arbitrarily larger than . Given a graph , let be the graph formed from by adding a new vertex and joining it to an vertex of degree one in . Then , , and . But since , and by Theorem 66. The gap between and is .

6. Results on Planar Graphs

From Section 2, we have seen that the bondage number for a tree has been completely solved. Moreover, a linear time algorithm to compute the bondage number of a tree is designed by Hartnell et al. [15]. It is quite natural to consider the bondage number for a planar graph. In this section, we state some results and problems on the bondage number for a planar graph.

6.1. Conjecture on Bondage Number

As mentioned in Section 3, the bondage number can be much larger than the maximum degree. But for a planar graph , the bondage number cannot exceed too much. It is clear that by Corollary 15 since for any planar graph . In 1998, Dunbar et al. [21] posed the following conjecture.

Conjecture 73. If is a planar graph, then .

Because of its attraction, it immediately became the focus of attention when this conjecture is proposed. In fact, the main aim concerning the research of the bondage number for a planar graph is focused on this conjecture.

It has been mentioned in Theorems 1 and 60 that , and . It is known that , where is a perfect matching of the complete graph . These examples show that if Conjecture 73 is true, then the upper bound is sharp for .

Here, we note that it is sufficient to prove this conjecture for connected planar graphs, since the bondage number of a disconnected graph is simply the minimum of the bondage numbers of its components.

The first paper attacking this conjecture is due to Kang and Yuan [45], which confirmed the conjecture for every connected planar graph with . The proofs are mainly based on Theorems 16 and 18.

Theorem 74 (Kang and Yuan [45], 2000). If is a connected planar graph, then .

Obviously, in view of Theorem 74, Conjecture 73 is true for any connected planar graph with .

6.2. Bounds Implied by Degree Conditions

As we have seen from Theorem 74, to attack Conjecture 73, we only need to consider connected planar graphs with maximum . Thus, studying the bondage number of planar graphs by degree conditions is of significance. The first result on bounds implied by degree conditions is obtained by Kang and Yuan [45].

Theorem 75 (Kang and Yuan [45], 2000). If is a connected planar graph without vertices of degree , then .

Fischermann et al. [46] generalized Theorem 75 as follows.

Theorem 76 (Fischermann et al. [46], 2003). Let be a connected planar graph and the set of vertices of degree 5 which have distance at least 3 to vertices of degrees 1, 2, and 3. If all vertices in not adjacent with vertices of degree are independent and not adjacent to vertices of degree , then .

Clearly, if has no vertices of degree 5, then . Thus, Theorem 76 yields Theorem 75.

Use to denote the number of vertices of degree in for each . Using Theorem 18, Fischermann et al. obtained the following two theorems.

Theorem 77 (Fischermann et al. [46], 2003). For any connected planar graph ,(1) if ;(2) if contains no vertices of degrees 4 and 5.

Theorem 78 (Fischermann et al. [46], 2003). For any connected planar graph , if(1) and every edge with and is contained in at most one triangle;(2) and no triangle contains an edge with and .

6.3. Bounds Implied by Girth Conditions

The girth of a graph is the length of the shortest cycle in . If has no cycles, we define .

Combining Theorem 74 with Theorem 78, we find that if a planar graph contains no triangles and has maximum degree , then Conjecture 73 holds. This fact motivated Fischermann et al.’s [46] attempt to attack Conjecture 73 by girth restraints.

Theorem 79 (Fischermann et al. [46], 2003). For any connected planar graph ,

The first result in Theorem 79 shows that Conjecture 73 is valid for all connected planar graphs with and . It is easy to verify that the second result in Theorem 79 implies that Conjecture 73 is valid for all not -regular graphs of girth , which is stated in the following corollary.

Corollary 80. For any connected planar graph , if is not -regular and , then .

The first result in Theorem 79 also implies that if is a connected planar graph with no cycles of length 3, then . Hou et al. [47] improved this result as follows.

Theorem 81 (Hou et al. [47], 2011). For any connected planar graph , if contains no cycles of length , then .

Since for (see Theorem 1), the last bound in Theorem 79 is tight. Whether other bounds in Theorem 79 are tight remains open. In 2003, Fischermann et al. [46] posed the following conjecture.

Conjecture 82. For any connected planar graph , , and

We conclude this subsection with a question on bondage numbers of planar graphs.

Question 1 (Fischermann et al. [46], 2003). Is there a planar graph with ?

In 2006, Carlson and Develin [42] showed that the corona for a planar graph with has the bondage number (see Theorem 66). Since the minimum degree of planar graphs is at most 5, then can attach 6. If we take as the graph of the icosahedron, then is such an example. The question for the existence of planar graphs with bondage number 7 or 8 remains open.

6.4. Comments on the Conjectures

Conjecture 73 is true for all connected planar graphs with minimum degree by Theorem 16, or maximum degree by Theorem 74, or not -critical planar graphs by Theorem 13. Thus, to attack Conjecture 73, we only need to consider connected critical planar graphs with degree-restriction .

Recalling and anatomizing the proofs of all results mentioned in the preceding subsections on the bondage number for connected planar graphs, we find that the proofs of these results strongly depend upon Theorem 16 or Theorem 18. In other words, a basic way used in the proofs is to find two vertices and with distance at most two in a considered planar graph such that which bounds , is as small as possible. Very recently, Huang and Xu [44] have considered the following parameter where and .

It follows from Theorems 16 and 18 that The proofs given in Theorems 74 and 79 indeed imply the following stronger results.

Theorem 83. If is a connected planar graph, then

Thus, using Theorem 16 or Theorem 18, if we can prove Conjectures 73 and 82, then we can prove the following statement.

Statement. If is a connected planar graph, then
It follows from (24) that Statement implies Conjectures 73 and 82. However, Huang and Xu [44] gave examples to show that none of conclusions in Statement is true. As a result, they stated the following conclusion.

Theorem 84 (Huang and Xu [44], 2012). It is not possible to prove Conjectures 73 and 82, if they are right, using Theorems 16 and 18.

Therefore, a new method is needed to prove or disprove these conjectures. At the present time, one cannot prove these conjectures, and may consider to disprove them. If one of these conjectures is invalid, then there exists a minimum counterexample with respect to . In order to obtain a minimum counterexample, one may consider two simple operations satisfying these requirements, the edge-deletion and the edge-contraction, which decrease and preserve planarity. However, applying Theorems 71 and 72, Huang and Xu [44] presented the following results.

Theorem 85 (Huang and Xu [44], 2012). It is not possible to construct minimum counterexamples to Conjectures 73 and 82 by the operation of an edge-deletion or an edge-contraction.

7. Results on Crossing Number Restraints

It is quite natural to generalize the known results on the bondage number for planar graphs to for more general graphs in terms of graph-theoretical parameters. In this section, we consider graphs with crossing number restraints.

The crossing number of is the smallest number of pairwise intersections of its edges when is drawn in the plane. If , then is a planar graph.

7.1. General Methods

Use to denote the number of vertices of degree in for . Huang and Xu obtained the following results.

Theorem 86 (Huang and Xu [48], 2007). For any connected graph , where ; ; ; and .

Simplifying the conditions in Theorem 86, we obtain the following corollaries immediately.

Corollary 87 (Huang and Xu [48], 2007). For any connected graph ,

Corollary 88 (Huang and Xu [48], 2007). For any connected graph ,(a) if is not -regular, and ;(b) if is not -regular, and ;(c) if is not -regular, and ;(d) if , and .

These corollaries generalize some known results for planar graphs. For example, Corollary 87 contains Theorem 79; Corollary 88 (b) contains Corollary 80.

Theorem 89 (Huang and Xu [48], 2007). For any connected graph , if , then .

Corollary 90 (Huang and Xu [48], 2007). For any connected graph , if , then .

Perhaps being unaware of this result, in 2010, Ma et al. [49] proved that for any graph with .

Theorem 91 (Huang and Xu [48], 2007). Let be a connected graph, and if , and for every . If is independent, no vertices adjacent to vertices of degree 6 and then .

Corollary 92 (Huang and Xu [48], 2007). Let be a connected graph with . If if and for every is independent, and no vertices adjacent to vertices of degree 6, then .

Theorem 93 (Huang and Xu [48], 2007). Let be a connected graph. If satisfies(a) or(b),then .

Proposition 94 (Huang and Xu [48], 2007). Let be a connected graph with no vertices of degrees four and five. If , then .

Theorem 95 (Huang and Xu [48], 2007). If is a connected graph with and not -regular when , then .

The above results generalize some known results for planar graphs. For example, Corollary 90 and Theorem 95 contain Theorem 74; the first condition in Theorem 93 contains the second condition in Theorem 77.

From Corollary 88 and Theorem 95, we suggest the following questions.

Question 2. Is there a(a)-regular graph with such that ?(b)-regular graph with and such that