The cost of edge removal in graph domination

A vertex set D of a graph G is a dominating set of G if each vertex of G is a member of D or is adjacent to a member of D . The domination number of G , denoted by c ð G Þ , is the cardinality of a smallest dominating set of G . In this paper two cost functions, d q ð G Þ and D q ð G Þ , are considered which measure respectively the smallest possible and the largest possible increase in the cardinality of a dominating set, over and above c ð G Þ , if q edges were to be removed from G . Bounds are established on d q ð G Þ and D q ð G Þ for a general graph G , after which these bounds are sharpened or these parameters are determined exactly for a number of special graph classes, including paths, cycles, complete bipartite graphs and complete graphs.

Let G ¼ ðV, EÞ be a simple graph of order n.A set D V is a dominating set of G if each vertex of G is a member of D or is adjacent to G. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by cðGÞ: Applications of the notion of domination abound: If the vertices of the graph G denote geographically dispersed facilities, and the edges model links between these facilities along which guards have line of sight, then a dominating set of G represents a collection of facility locations at which guards may be placed so that the entire complex of facilities modelled by G is protected (in the sense that if a security problem were to occur at facility u, there will either be a guard at that facility who can deal with the problem, or else a guard dealing with the problem from an adjacent facility v can signal an alarm due the visibility that exists between adjacent locations).In this application, the domination number represents the minimum number of guards required to protect the facility complex.

Edge removal
In applications conforming to the scenario described above one might seek the cost (in terms of the additional number of guards required over and above the minimum cðGÞ to protect an entire location complex G in the dominating sense) if a number of edges of G were to "fail" (i.e. a number of links were to be eliminated from the graph so that the guards no longer have vision along such disabled links).
In this paper, the notation Gqe is used to denote the set of all non-isomorphic graphs obtained by removing 0 q m edges from a given graph G of size m.Furthermore, cðG À qeÞ denotes the set of values of cðHÞ as H 2 G À qe varies (for a fixed value of q).Walikar and Acharya [7, Proposition 2] were the first to note the following result.
Proposition 1.Let G be any graph and e any edge of G. Then it follows that cðGÞ cðG À eÞ cðGÞ þ 1: The following result follows immediately from Proposition 1.
Corollary 1 (Edge removal increases domination requirements).For any graph G that is not edgeless cðGÞ min cðG À eÞ max cðG À eÞ cðGÞ þ 1: w The cost functions d q ðGÞ ¼ min cðG À qeÞ À cðGÞ D q ðGÞ ¼ max cðG À qeÞ À cðGÞ are non-negative in view of Corollary 1 and measure respectively the smallest possible and the largest possible increase in the minimum number of guards required to dominate a member of Gqe, over and above the minimum number of guards required to dominate G, in the event that an arbitrary set of 0 q m edges are removed from G. Furthermore, cost sequences dðGÞ ¼ d 0 ðGÞ, d 1 ðGÞ, d 2 ðGÞ, :::, d m ðGÞ and DðGÞ ¼ D 0 ðGÞ, D 1 ðGÞ, D 2 ðGÞ, :::, D m ðGÞ can be constructed for any graph G.
The cost functions d q ðGÞ and D q ðGÞ were first introduced by Burger et al. [2] for the domination related parameter secure domination.For a graph G with secure domination number c s ðGÞ it follows that cðGÞ c s ðGÞ [3, Proposition 1].
Van Vuuren [6] studied the notion of q-criticality in a graph G.A graph G is q-critical if q is the smallest number of arbitrary edges of G whose removal from G necessarily increases the domination number of the resulting graph.In this paper the cost sequence dðGÞ consequently produces the qcriticality of a graph G when d q ðGÞ > 0, but d qÀ1 ðGÞ ¼ 0: The notion of q-criticality have also been studied for other related graph parameters such as secure domination [4].
Proposition 2 (Cost function q-growth properties).If G is a graph of size m and 0 q < m, then (a) d q ðGÞ d qþ1 ðGÞ d q ðGÞ þ 1, and (b) D q ðGÞ D qþ1 ðGÞ D q ðGÞ þ 1: Proof: (a) By applying the result of Proposition 1 to each element of Gqe, it follows that which establishes the first inequality.The second inequality holds because the domination number of a graph cannot increase by more than 1 if a single edge is removed from the graph by Proposition 1.The proof of part (b) is similar. w The cost functions d q ðP 6 Þ and D q ðP 6 Þ are evaluated in Table 1 for the path P 6 of order 6 for all 0 q 5: (These results may be verified by recalling from [3,Theorem 12

General bounds on the cost sequences
The following general bounds hold with respect to the sequences dðGÞ and DðGÞ for any graph G.
Theorem 1.For any graph G of order n and size m, n À m þ q À aðGÞ d q ðGÞ D q ðGÞ q: Proof: It follows by Berge [ Finally, by applying the result of Proposition 2(b) q times, it follows that D q ðGÞ q: w The bounds in Theorem 1 are sharp; they are attained by taking G to be the vertex disjoint union of paths of order 1 and 2 (in which case aðGÞ ¼ n À m).

Special graph classes
In this section exact values of or bounds on the sequences dðGÞ and DðGÞ are established for a number of special classes of graphs, including paths, cycles, complete bipartite graphs and complete graphs.

Paths and cycles
In this section P n and C n denote a path and a cycle of order n, respectively.It follows by Theorem 1 that for all n ! 2 and 0 q n À 1, by noting that aðP n Þ ¼ d n 2 e: However, these bounds are weak, especially for small values of q.In this section the sequences dðP n Þ and DðP n Þ are determined exactly and these results are used to derive the sequences dðC n Þ and DðC n Þ: For this purpose the following basic result is required.

Lemma 1.
(a) For n ! 4 and any Proof: (a) Suppose n ! 4 and let k be any positive integer not exceeding n -1.Then Table 1.The costs d q ðP 6 Þ and D q ðP 6 Þ for the path P 6 .
by means of the identity dae þ db À ae !dbe for any a, b 2 R: (b) Suppose n ! 5 and let k be any positive integer not exceeding n -1.Then by (three times) using the identity The following intermediate results are also required.
Lemma 2. Suppose E, F 2 P n À qe respectively minimise and maximise cðP n À qeÞ: Proof: (a) By contradiction.Suppose 2q n 3q and that which contradicts the supposition that n 3q: (b) By contradiction.Suppose 3q < n and that G 2 P nþ3 À ðq þ 1Þe minimises cðP nþ3 À ðq þ 1ÞeÞ, but that cðGÞ < cðE [ P 3 Þ: Then G contains no component of order 3 and it follows by Lemma 1(a) that no two components of G together have more than three vertices.It is therefore assumed that G ffi xP 2 [ yP 1 : By evaluating the number of components and the number of vertices of G, it follows that x þ y ¼ q þ 2 and 2x þ y ¼ n þ 3, respectively.The unique solution to this simultaneous system of equations is x ¼ n À q þ 1 and y ¼ 2q À n þ 1: Since y !0 it follows that 2q !n À 1, contradicting the supposition.(c) By contradiction.Suppose n À 3 q n À 1 and that H 2 P nþ1 À ðq þ 1Þe maximises cðP nþ1 À ðq þ 1ÞeÞ, but that cðHÞ > cðF [ P 1 Þ: Then H is isolate-free and dðHÞ !2: But then the order of H is n þ 1 > 2ðq þ 2Þ, which contradicts the supposition that n q þ 3: (d) By contradiction.Suppose q < n À 3 and that H 2 P nþ4 À ðq þ 1Þe maximises cðP nþ4 À ðq þ 1ÞeÞ, but that cðHÞ > cðF [ P 4 Þ: Then H contains no component of order 4 and it follows by Lemma 1(b) that no two components of H together have more than four vertices.Furthermore, the equality show that there is at least one member of P nþ4 À ðq þ 1Þe which maximises cðP nþ4 À ðq þ 1ÞeÞ and which has at most one component which is not an isolate.It is therefore assumed that G ffi P i [ xP 1 for some i 2 f2, 3g: By evaluating the number of components and the number of vertices of H, it follows that respectively, which together imply that n ¼ q þ i À 3: However, this equality contradicts the supposition that q < n À 3 for i ¼ 2, 3. Suppose n 2 N and q 2 N 0 such that q n À 1. Then Proof: Both cases of the formula above for d q ðP n Þ are established by means of induction over q.Suppose n > 3q, for which the base case is d 0 ðP n Þ ¼ 0 and that E n 2 P n À 'e minimises cðP n À 'eÞ: Assume, as induction hypothesis that the desired formula holds for q ¼ ', i.e. min cðP n À 'eÞ f g¼ d n 3 e for all ' < n 3 : To show that the formula also holds for q ¼ ' þ 1, a disjoint path P 3 is added to E n for all n > 3': Then it follows by Lemma 2(b) that and thereby completing the induction process for this case.Suppose next that n 3q and suppose that E n 2 P n À 'e minimises cðP n À 'eÞ and assume, as induction hypothesis, that the formula holds for q ¼ ', i.e. minfcðP n À 'eÞg ¼ ' þ 1 for all n 3': To show that the formula also holds for q ¼ ' þ 1 a disjoint path P 2 is added to E n for 2' n 3', thereby covering the required range of values of n for q thereby completing the induction process for 2' n 3': Finally, suppose n < 2q and consider d 2 ðP 3 Þ ¼ 2 as base case.Assume, as induction hypothesis, that the formula holds for q ¼ ', i.e. minfcðP n À 'eÞg ¼ q þ 1 for n < 3': Let E n 2 P n À 'e and suppose the vertex set of E n is fv 1 , :::, v n g: It is shown by contradiction that E n has at least one isolated vertex.Assume, to the contrary, that E n has no isolated vertex.Then it follows by the handshaking lemma that n since each vertex has degree at least one.Therefore, n 2ðn À 1 À 'Þ, or equivalently n !2' þ 2, which contradicts the fact that n < 2' þ 2: Hence, E n has at least one isolated vertex, and so thereby completing the induction process.
The formula above for D q ðP n Þ are established by induction over q and suppose that q < n À 3 and suppose that F n 2 P n À 'e maximises cðP n À 'eÞ and assume, as induction hypothesis, that the formula holds for q ¼ ', i.e. max cðP n À 'eÞ f g¼d nþ2' 3 e for all ' < n À 3: To show that the formula also holds for q ¼ ' þ 1, a disjoint path P 4 is added to F n for q < n À 3, thereby covering the required range of values of n for q ¼ ' þ 1, i.e. ' þ 4 < n À 3: Then it follows by Lemma 2(d) that thereby completing the induction process for ' < n À 3: Suppose next that n À 3 ' n À 1 and suppose that F n 2 P n À 'e maximises cðP n À 'eÞ: Assume, as induction hypothesis, that the formula holds for q ¼ ', i.e. max cðP n À 'eÞ f g¼ d nþ2' 3 e for all n À 3 ' n À 1: To show that the formula also holds for q ¼ ' þ 1, a disjoint path P 1 is added to F n for n À 3 ' n À 1, thereby covering the required range of values of n for q thereby completing the induction process for n À 3 ' n À 1: w The next result immediately follows from Theorem 2, because C n À e contains a single element, which is isomorphic to P n , for all n !3: Corollary 2 (The sequences d and D for cycles) Suppose n 2 N and q 2 N 0 such that q n.Then

Complete bipartite graphs
It follows by Theorem 1 that n À ðj þ 1Þðn À jÞ þ q d q ðK j, nÀj Þ D q ðK j, nÀj Þ q for all n À j !j and 0 q jðn À jÞ, by noting that aðK j, nÀj Þ ¼ n À j: Again, these bounds seem to be weak for small values of q.
For the simplest class of complete bipartite graphs, namely stars, it is possible to determine the values of d and D exactly.For the simplest class of complete bipartite graphs, namely stars, it holds that Perhaps the most simple and most natural generalisation of a star, namely the graph K 2, nÀ2 , is considered.Theorem 3.For the complete bipartite graph K 2, nÀ2 of order n ! 4, Proof: Denote the partite sets of K 2, nÀ2 by fx, yg and V ¼ fv 1 , :::, v nÀ2 g: Removing q edges from K 2, nÀ2 results in a subgraph G 2 K 2, nÀ2 À qe ¼: Kðn, qÞ and the partition V ¼ x (V G y , respectively) contains the vertices adjacent to x only (y only, resp.) in G, and V G xy contains the common neighbours of x and y in G.
In order to determine a minimum dominating set for G, two mutually exclusive cases are considered.If 0 q n À 2, then the number of vertices in V G 0 is minimised by removing from K 2, nÀ2 the edges xv 1 , xv 2 , xv 3 , and so on, in this order, until q edges have been removed.In this way, jV G 0 j ¼ jV G x j ¼ 0, jV G y j ¼ q and jV G xy j ¼ n À q À 2, resulting in the expression as in Case i and Case ii (b).If n À 2 < n 2n À 4, then the number of vertices in V G 0 is minimised by removing the edges xv 1 , xv 2 , :::, xv nÀ2 together with the edges yv 1 , yv 2 , yv 3 , and so on, in this order, until q edges have been removed.In this way, jV G 0 j ¼ q À ðn À 2Þ, jV G x j ¼ 0, jV G y j ¼ ð2n À 4Þ À q and jV G xy j ¼ 0, resulting in the expression as in Case i and Case ii (b).
The number of vertices in V G 0 is maximised by removing from K 2, nÀ2 the edges xv 1 , yv 1 , xv 2 , yv 2 , xv 3 , yv 3 , and so on, in this order, until q edges have been removed.In this way, as in Case i.If 2ðn À 4Þ < q 2ðn À 2Þ, then the number of vertices in V G 0 is maximised by removing from K 2, nÀ2 the edges xv 1 , yv 1 , xv 2 , yv 2 , xv 3 , yv 3 , and so on, in this order, until 2n À 8 edges have been removed.It follows that xy , then the vertices fx, y, z 1 , z 2 g induce a cycle of order four, yielding the result due to the result from Corollary 2 in conjunction with Case ii (a) and (b). w From the results of Theorem 3 it is possible to generalise the result for the graph K j, nÀj , where j > 2 for the cost function d q ðK j, nÀj Þ: This process is simplified by the realisation that cðK j, nÀj Þ ¼ 2 for all n À j !j and j !3: A simple sequence of edge removals can be shown to provide an exact formulation for d q ðK j, nÀj Þ: Theorem 4. For the complete bipartite graph K j, nÀj of order n À j !j ! 3, then Denote the partite sets of K j, nÀj by X ¼ fx 1 , :::, x j g and Y ¼ fy 1 , :::, y nÀj g: The set fx 1 , y 1 g is a minimum dominating set for K j, nÀj by Cockayne et al. [3,Proposition 10(a)].Removing q edges from K j, nÀj results in a subgraph G 2 K j, nÀj ¼: K 0 ðn, qÞ and the denote E G x 0 1 as the set of edges incident to y k for k ¼ 2, 3, :::, n À j, and similarly, denote E G y 0 1 as the set of edges incident with x ' for ' ¼ 2, 3, :::, j: If 0 q ðj À 1Þðn À j À 1Þ þ 1 the number of edges incident with the dominating set fx 1 , y 1 g are not to be removed.The removal of any edge from the edge set where k !' ! 2 yields a subgraph of K j, nÀj for which fx 1 , y 1 g is a dominating set of K j, nÀj À qe: By Case i, it follows that For ðj À 1Þðn À j À 1Þ þ 2 q jðn À jÞ, the removal of the edges x ' y k for k ¼ 2, :::, n À j and ' ¼ 2, :::, j and finally the edge x 1 y 1 yields two disjoint stars, K 1, nÀj and K 1, j with universal vertices x 1 and y 1 , respectively.It follows that the removal of any subsequent edge from K j, nÀj increases the domination number, and as a result it holds that d q ðK j, nÀj Þ ¼ min G2K 0 ðn, qÞ fcðGÞg À 2 ¼ q À j 0 , if j 0 < q jðn À jÞ, by Case ii where j 0 ¼ ðj À 1Þðn À j À 1Þ þ 1: w It seems rather difficult to generalise the result of Theorem 3 for the cost function D q ðK j, nÀj Þ where j > 2, because of the large number of cases involved in a generalisation of the proof in Theorem 3. It is however possible to provide an algorithmic lower bound for D q ðK j, nÀj Þ where j > 2. A pseudo-code listing of this iterative procedure is given in the guise of a breadth-first search as Algorithm 1.The algorithm is based on the principle of iteratively isolating vertices of largest degree until the empty graph remains.The bounding sequence in Algorithm 1is expected to be good approximations of the sequences DðK j, nÀj Þ: The algorithm maintains a list DBoundSequence: This list is populated with appropriate lower bounds on D q ðGÞ for a graph G during execution of the algorithm.For example, for the graph K 3, 5 the list DBoundSequence is 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 6:

It follows by Theorem
Þ q, but these bounds are weak for small q.
Theorem 5.For the complete graph K n of order n, it follows that Case ii: jE G x j ¼ 0 and jE G x j n À 1: In this case G is the vertex disjoint union of the isolated vertices, say V G 0 , and a star with universal vertex fxg.Therefore G is dominated by the vertices in V G 0 [ fxg, and no smaller dominating set of G exists by Cockayne et al. [3,Proposition 10(a)].
Then it follows by Case i that fcðGÞg À 1 ¼ 0, if 0 < q n À 1 2 : , the removal of the edges E G x , yields a star K 1, nÀ1 with x as universal vertex.Any subsequent edge removal increases the domination number of G and as a result follows holds that , by Case ii.
w Again Algorithm 1 is considered to aid in providing a lower bound on D q ðK n Þ: For the graph K 6 , the list DBoundSequence is 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5: It is important to note that Algorithm 1 is not a suitable approximation of D q ðGÞ for any graph G in general.Special graph classes such as the complete bipartite graph K j, nÀj and complete graph K n of orders n are suited candidates as input for Algorithm 1.However, it remains an open problem whether Algorithm 1 does provide the exact cost sequence D for complete graphs and complete bipartite graphs.

Conclusions
In this paper, two cost function sequences, dðGÞ and DðGÞ for a graph G were introduced and illustrated in §2.These sequences measure respectively the smallest and largest increase of cðGÞ as edges are removed from G. General bounds on dðGÞ and DðGÞ were established in §3, after which exact values for or bounds on these functions were determined in §4 for a number of special graph classes, including, paths, cycles, complete bipartite graphs and complete graphs.
Further, related work may include determining the value of c for other graph classes, such as complete multipartite graphs, trees, circulant graphs and various Cartesian products.Furthermore, exact formulations on the cost sequence D for complete graphs and complete bipartite graphs remains open for further research.

w
It is now possible to establish the sequences d and D for paths.Theorem 2 (The sequences d and D for paths) Case i: jV G xy j 6 ¼ 1: In this case G is dominated by the vertices in V G 0 [ fx, yg, and no smaller dominating set of G exists by Cockayne et al. [3, Proposition 10(a)].Case ii (a): jV G xy j ¼ 1 and jV G x j ¼ jV G y j ¼ 0: In this case G is the vertex disjoint union of the isolated vertices in V G 0 and a star with universal vertex fzg 2 V G xy : Therefore G is dominated by the vertices in V G 0 [ fzg, and no smaller dominating set of G exists by Cockayne et al. [3, Proposition 10(a)].Case ii (b): jV G xy j ¼ 1, jV G x j > 0 and jV G y j > 0: In this case G is again dominated by the vertices in V G 0 [ fx, yg, and no smaller dominating set of G exists by Cockayne et al. [3, Proposition 10(a)].

þ 1 : 1 j ¼ n À j À 1 and jE G x 0 1 j ¼ j À 1 : 1 j n À j À 1 and jE G x 0 1 j j À 1 :
In order to determine a minimum dominating set for G, two mutually exclusive cases are considered.Case i: jE G rem j !0 and jE G x 0 In this case G is dominated by the vertex in fx 1 , y 1 g, and no smaller dominating set of G exists by Cockayne et al. [3, Proposition 10(a)].Case ii: jE G rem j ¼ 0 and jE G x 0 In this case G is the vertex disjoint union of the isolated vertices, say V G 0 , and two disjoint stars with universal vertices x 1 and y 1 , respectively.Therefore G is dominated by the vertices in V G 0 [ fx 1 , y 1 g, and no smaller dominating set of G exists by Cockayne et al. [3, Proposition 10(a)].
Algorithm 1: A lower bound on the sequence D for K n or K j, nÀj Input: The complete graph K n or the complete bipartite graph K j, nÀj of order n.
where E G x are the edges incident with the vertex x, and E G x are the set of edges incident with the vertex set VðK n Þ fxg: In order to determine a minimum dominating set for G, two mutually exclusive cases are considered.Case i: jE G x j 6 ¼ 0 and jE G x j ¼ n À 1: In this case G is dominated by the vertex in fxg, and no smaller dominating set of G exists by Cockayne et al. [3, Proposition 10(a)].