Upper Bounds on the Bondage Number of a Graph

The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. We obtain sufficient conditions for the validity of the inequality $b(G) \leq 2s - 2$, provided $G$ has degree s vertices. We also present upper bounds for the bondage number of graphs in terms of the girth, domination number and Euler characteristic. As a corollary we give a stronger bound than the known constant upper bounds for the bondage number of graphs having domination number at least four. Several unanswered questions are posed.


Introduction
An orientable compact 2-manifold S h or orientable surface S h (see [21]) of genus h is obtained from the sphere by adding h handles. Correspondingly, a non-orientable compact 2-manifold N q or non-orientable surface N q of genus q is obtained from the sphere by adding q crosscaps. Compact 2-manifolds are called simply surfaces throughout the paper. The Euler characteristic is defined by χ(S h ) = 2 − 2h, h ≥ 0, and χ(N q ) = 2 − q, q ≥ 1. The Euclidean plane S 0 , the projective plane N 1 , the torus S 1 , and the Klein bottle N 2 are all the surfaces of non-negative Euler characteristic.
We shall consider graphs without loops and multiple edges. A graph G is embeddable on a topological surface M if it admits a drawing on the surface with no crossing edges. Such a drawing of G on the surface M is called an embedding of G on M. If a graph G is embedded in a surface M then the connected components of M − G are called the faces of G. For such a graph G, we denote its vertex set, edge set, face set, maximum degree, and minimum degree by V (G), E(G), F (G), ∆(G), and δ(G), respectively. Set |G| = |V (G)|, G = |E(G)|, and f (G) = |F (G)|. An embedding of a graph G on a surface M is said to be 2-cell if every face of the embedding is homeomorphic to an open disc. The Euler's inequality states for any graph G that is embedded in M. Equality holds if G is 2-cell embedded in M. By the genus h (the non-orientable genus q) of a graph G we mean the smallest integer h (q) such that G has an embedding into S h (N q , respectively).
The girth of a graph G, denoted as g(G), is the length of a shortest cycle in G; if G is a forest then g(G) = ∞. The distance between two vertices x, y ∈ V (G) is denoted by d G (x, y). The average degree ad(G) of a graph G is defined as ad(G) = 2 G /|G|.
An independent set is a set of vertices in a graph, no two of which are adjacent. The independence number β 0 (G) of a graph G is the size of the largest independent set in G. A dominating set for a graph G is a subset D ⊆ V (G) of vertices such that every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set is called the domination number of G and is denoted by γ(G). The concept of domination in graphs has many applications in a wide range of areas within the natural and social sciences. One measure of the stability of the domination number of G under edge removal is the bondage number b(G) defined in [2] (previously called the domination line-stability in [2]) as the smallest number of edges whose removal from G results in a graph with larger domination number. We refer the reader to [31] for a detailed survey on this topic. In general it is N P -hard to determine the bondage number (see Hu and Xu [11]), and thus useful to find bounds for it.
Hartnell and Rall [8] and Teschner [30] showed that for the Cartesian product G n = K n × K n , n ≥ 2, the bound of Conjecture 1 is sharp, i.e. b(G n ) = 3 2 ∆(G n ). Teschner [29] also proved that Conjecture 1 holds when the domination number of G is not more than 3.
The study of the bondage number of graphs, which are 2-cell embeddable on a surface having negative Euler characteristic was initiated by Gagarin and Zverovich [6] and is continued by the same authors in [7], Jia Huang in [12] and the present author in [24]. All these authors obtain upper bounds for the bondage number in terms of maximum degree and/or orientable and non-orientable genus of a graph. In [25], the present author gives upper bounds for the bondage number in terms of order, girth and Euler characteristic of a graph. By Theorem 10 (ii) [7] or by Theorem B(ii) below, it immediately follows that Conjecture 1 is true for any graph G such that all the following is valid: (a) G is 2-cell embeddable in a surface M with χ(M) < 0, (b) |G| > −12χ(M), and (c) ∆(G) ≥ 8.
In this paper we concentrate mainly on the case when a graph G is 2-cell embeddable in a surface M and |G| ≤ −12χ(M). The rest of the paper is organized as follows. Section 2 contains preliminary results. In section 3 we give new arguments that improve the known upper bounds on the bondage number at least when −7χ(M)/(δ(G) − 5) < |G| ≤ −12χ(M), δ(G) ≥ 6. We propose a new type of upper bound on the bondage number of a graph. Namely we obtain sufficient conditions for the validity of the inequality b(G) ≤ 2s − 2, where G is a graph having degree s vertices, s ≥ 5. In particular, we prove that if a connected graph G is 2-cell embeddable in an orientable/non-orientable surface M with negative Euler characteristic then b(G) ≤ 2δ − 2 whenever −14χ(M) < δ(G) − 4 + 2(δ(G) − 5)|G| and δ(G) ≥ 6. We also improve the known upper bounds for b(G) when a graph G is embeddable on at least one of N 1 , N 2 , N 3 , N 4 and S 2 . In section 4 we give tight lower bounds for the number of vertices of graphs in terms of Euler characteristic and the domination number. We also present upper bounds for the bondage number of graphs in terms of the girth, domination number and Euler characteristic. As a corollary, in section 5 we give a stronger bound than the known constant upper bounds for the bondage number of graphs having domination number at least 4.

Known and preliminary results
In this section we recall several known upper bounds on the bondage number of a graph and prove some useful lemmas. We need the following notations and definitions. • Theorem A. If G is a nontrivial graph, then (i) (Hartnell and Rall [9] (ii) (Hartnell and Rall [8]

By Theorem A and the above definitions we have
Note that, if a graph G has no triangles then B(G) = B ′ (G) = b 1 (G).
Theorem B. (Samodivkin [25]). Let G be a connected graph embeddable on a surface M whose Euler characteristic χ is as large as possible and let g(G) = g. If χ ≤ −1 then: The same upper bound for b(G), in case when g ∈ {3, 4}, is obtained by Gagarin and Zverovich [6].
Theorem C. (Gagarin and Zverovich [7]). Let G be a connected graph 2-cell embedded in a surface M with Theorem D. (Samodivkin [24]). Let G be a connected toroidal or Klein bottle graph. Then b 2 (G) ≤ ∆(G) + 3 with equality if and only if one of the following conditions is valid: (P4) G is 6-regular and no edge of G belongs to at least 3 triangles.
In [5], Frucht and Harary define the corona of two graphs G 1 and G 2 to be the graph G = G 1 • G 2 formed from one copy of G 1 and |G 1 | copies of G 2 , where the ith vertex of G 1 is adjacent to every vertex in the ith copy of G 2 .

Theorem E. (Carlson and Develin [3]). Let G be a graph of the form
g G , and the result easily follows.
The next lemma is fairly obvious and hence we omit the proof.
and the following hold: (a)-(c): The results immediately follow by the very definition of the graph G k and by Claim 1.

Upper bounds: degree s vertices
Motivated by Theorems A, B and C, in this section we concentrate on the set of all vertices of degree at most s in a 2-cell embedded graph, s ≥ 4. We impose some restrictions on this set to obtain new upper bounds on the bondage number. The main result of this section is the following theorem: Proof. Let G be a connected graph 2-cell embedded in a surface M with χ(M) = χ. Suppose B ′ (G) ≥ 2s − 1. Keeping the notation of Lemma 2.3 let us consider the graph H = G k − V ≤s−1 . By Clam 1 and Lemma 2.3 we immediately have: (a) δ(H) = s, I s = V s (H) and I s is an independent set of H.
By Lemma 2.1 and Claim 2 it follows that Let us consider the bipartite graph R with parts I s and N H (I s ), and edge set {uv ∈ E(G) | u ∈ I s , v ∈ N G (I s )}. First let R have a cycle. Lemma 2.1 implies s|I s | = R ≤ 2(|R| − χ). Since |R| = |I s | + |N H (I s )|, we obtain By (3) and (4) it follows Since |H| = |G| − |V ≤s−1 |, we finally obtain a contradiction.
The next two corollaries immediately follow from Theorem 3.1.
This solves Conjecture 1 when (a) G is as in Corollary 3.1(i) and ∆(G) ≥ 6 or (b) G is as in Corollary 3.1(ii) and ∆(G) ≥ 7 .
Hence we may conclude that Conjecture 1 is true whenever G is as in Corollary 3.2 and 4δ(G)− 4 ≤ 3∆(G).
< |G| ≤ −12χ then the bound stated in Corollary 3.2 is better than that given in Theorem B(ii). Proof. (a) Since χ(M) is as large as possible, G has 2-cell embedding on M [18]. Since G has no vertex of degree s = δ M max , V ≤s−1 is not empty. Suppose to the contrary that B ′ (G) ≥ 2s − 2. Hence, for any two distinct vertices x, y ∈ V ≤s−1 = {x 1 , . . . , x k }, d G (x, y) ≥ 3. Now, as in the proof of Lemma 2.3, we obtain a supergraph G k for G with V (G) = V (G k ) and xy ∈ E(G k ) − E(G) implies both x and y are in N G (u) for some u ∈ V ≤s (G). Moreover, if d G (x r ) ≥ 3 then N G k (x r ), G k is Hamiltonian, and if d G (x r ) = 2 then x r belongs to a triangle of G k , r = 1, 2, . . . , k.
If d G (u) = 2 and the equality holds then Consider the graph H = G k − V ≤s (G) which is embedded in M. Since s ≥ 5, by Claim 3 it follows δ(H) ≥ s + 1 -a contradiction.
(b) The result immediately follows by (a) and Lemma F.
There are infinitely many planar graphs G without degree δ S 0 max = 5 vertices for which B ′ (G) = 2δ S 0 max − 3 = 7. One such a graph is depicted in Figure 1. Notice that for a planar graph G without degree 5 vertices, the inequalities b(G) ≤ 7 and B(G) ≤ 7 are due to Kang and Yuan [16] and Huang and Xu [13], respectively. By Theorem 3.2 and Corollary 3.1(i) it immediately follows: The inequalities b(G) ≤ 8 and B(G) ≤ 8 for planar graphs, were proven by Kang and Yuan [16] and Huang and Xu [13], respectively. Consider the planar graph H shown in Figure 2  Carlson and Develin [3] showed that there exist planar graphs with bondage number 6. It is not known whether there is a planar graph G with b(G) ∈ {7, 8}.
Consider the projective-planar graph R depicted in Figure 3. Note that R is a triangulation, each edge of R is in exactly 2 triangles, δ(R) = 5, there are no adjacent degree 5 (red) vertices and there is a degree 5 vertex adjacent to a degree 6 (black) vertex. This implies B(R) = B ′ (R) = 8. Hence the upper bound for B ′ (G) in Corollary 3.3 is tight when M = N 1 . Note that in the case when M = N 1 , our result is better than b(G) ≤ 10 which was recently and independently obtained by Gagarin and Zverovich [7] and by the present author [25]. It is well known that the non-orientable genus of K 6 is 1 [21]. Hence by Theorem E we obtain: Proposition 3.1. There exist projective-planar graphs with bondage number 6. In particular, b(K 6 • K 1 ) = 6. In the next corollary we improve the known upper bound for the bondage number of Klein bottle graphs from 11 (Gagarin and Zverovich [7]) to 9. Proof. If δ(G) ≥ 6 then G is a 6-regular triangulation as it follows by the Euler formula; hence B ′ (G) = 9. If V 5 (G) is not empty then B ′ (G) ≤ 8 by Corollary 3.1. So, let V ≤4 (G) = ∅ and is a graph with minimum degree at least 6 and maximum degree at least 7, contradicting Lemma 2.1. Consider the supergraph G k of G described in Lemma 2.3, provided s = 5. Then Lemma 2.3 implies the graph H = G k − V ≤4 has minimum degree at least 6 and maximum degree at least 7 -again a contradiction with Lemma 2.1.
It is an immediate consequence of Euler's formula that any 6-regular graph embedded in M ∈ {S 1 , N 2 } is a triangulation. Altshuler [1] found a characterization of 6-regular toroidal graphs and Negami [19] characterized 6-regular graphs which embed in the Klein bottle. Moreover, no 6-regular graph embeds in both the torus and the Klein bottle [17]. The inequality b(G) ≤ 9 for toroidal graphs, was proven by Hou and Liu [10]. They also showed that there exist toroidal graphs with bondage number 7. The next result immediately follows by Theorem E.  (ii) If µ(G) = 3 2 ∆(G) then either G is 6-regular and no edge of G belongs to at least 3 triangles or 3 ≤ δ(G) ≤ ∆(G) = 4.
We improve this bound in the following corollary. Proof. If G is embeddable in a surface with non-negative Euler characteristic then the result follows by Corollary 3.3 and Corollary 3.4. So, we may assume that the non-orientable genus of G is 3 and hence |G| ≥ 7. By Lemma 2.1, G ≤ 3|G| + 3. Hence δ N 3 max = 6. If G has no degree 6 vertices then B ′ (G) ≤ 9 because of Theorem 3.2. Assume V 6 is not empty. But then 7 < β 0 ( V 6 , G ) + |G|. Now by Corollary 3.1(ii), b(G) ≤ B ′ (G) ≤ 10. The rest immediately follows by Corollary 3.1(i).
Since the non-orientable genus of K 7 is 3 [21], by Theorem E we obtain: Proposition 3.4. There exist graphs embeddable on N 3 with bondage number 7. One of them is Question. Is there a graph G embeddable in N 3 with b(G) ∈ {8, 9, 10}?
We conclude our results in this section with a constant upper bound on the bondage number of graphs embeddable in M ∈ {S 2 , N 4 }. For any such a graph G, b(G) ≤ 16 (Gagarin and Zverovich [7]). We improve this result as follows.

Upper bounds: the domination number
In this section (a) we present upper bounds for the order of a graph in terms of the domination number and Euler characteristic, and (b) we give upper bounds for the bondage number in terms of the girth, domination number and Euler characteristic. The obtained bounds for b(G) are better than the one in Theorem C. We need the following results.
Theorem G. (Sanchis [26]) Let G be a connected graph with n vertices and domination number γ where 3 ≤ γ ≤ n/2. Then the number of edges of G is at most (n − γ + 1)(n − γ)/2. If G has exactly this number of edges and γ ≥ 4 it must be of the following form.
(P 1 ) An (n − γ)-clique, together with an independent set of size γ, such that each of the vertices in the (n − γ)-clique is adjacent to exactly one of the vertices in the independent set, and such that each of these γ vertices has at least one vertex adjacent to it.
(P 2 ) For γ = 3, G may consist of a clique of n−5 vertices, together with 5 vertices x 1 , x 2 , x 3 , x 4 , x 5 , with edges x 1 x 3 , x 2 x 4 , x 2 x 5 , such that every vertex in the (n − 5)-clique is adjacent to x 4 and x 5 , and in addition adjacent to either x l or x 3 . Moreover, at least one of these vertices is adjacent to x l and at least one to x 3 .
Combining Theorem B(i) and Proposition 4.1 we immediately obtain the following results on the average degree of a graph.

Remarks
Teschner [29] proved that Conjecture 1 holds when the domination number of a graph G is not more than 3.
Hence it is naturally to turn our attention toward the graphs with the domination number at least 4. By  Table 1 provided γ(G) ≥ 4.  For the sake of completeness we add the upper bounds presented in section 3.