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Changing and Unchanging in m-Eternal Total Domination in Graphs
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Electronic Notes in Discrete Mathematics 53 (2016) 181–198 www.elsevier.com/locate/endm
Changing and Unchanging in m-Eternal Total Domination in Graphs P. Roushini Leely Pushpam1 Department of Mathematics D.B. Jain College Chennai 600 097, Tamil Nadu, India
PA. Shanthi2 Department of Mathematics Sri Sairam Engineering College Chennai 600 044, Tamil Nadu, India Abstract The Eternal dominating set of a graph is defined as a set of guards located at vertices, required to protect the vertices of the graph against infinitely long sequences of attacks, such that the configuration of guards induces a dominating set at all times. The eternal m-security number is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. Klostermeyer and Mynhardt introduced the concept of an m-eternal total dominating set, which further requires that each vertex with a guard to be adjacent to a vertex with a guard. They defined the m-eternal total domination number of a graph G denoted ∞ (G) as the minimum number of guards to handle an arbitrary sequence of by γmt single attacks using multiple guard shifts and the configuration of guards always ∞ (G) induces a total dominating set. In this paper we examine the effects on γmt when G is modified by deleting a vertex which is a non support vertex. Keywords: Eternal total domination, total domination. 1 2
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http://dx.doi.org/10.1016/j.endm.2016.05.017 1571-0653/© 2016 Elsevier B.V. All rights reserved.
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1
Introduction
Let G = (V, E) be a simple and connected graph of order |V | = n. For graph theoretic terminology, we refer to Harary [7]. For any vertex v ∈ V , the open neighbourhood of v is the set N (v) = {u ∈ V |uv ∈ E} and the closed neighbourhood is the set N [v] = N (v) ∪ {v}. For a set S ⊆ V , the open neighbourhood is N (S) = ∪v∈s N (v) and the closed neighbourhood is N [S] = N (S) ∪ S. A set S is a dominating set, if N [S] = V (G) or equivalently, every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G, and a dominating set S of minimum cardinality is called a γ-set of G. A total dominating set (TDS) of G is a set D ⊆ V with the property that for each u ∈ V , there exists x ∈ D adjacent to u. The minimum cardinality amongst all total dominating sets is the total domination number γt (G). A TDS S of minimum cardinality is called γt -set of G. It can be noted that this parameter is only defined for graphs without isolated vertices. Burger et al. [2,3] introduced a dynamic form of domination which has been designated eternal security by Goddard et al. [6]. In this concept, fixed number of guards are positioned on the vertices of a graph G = (V, E), at most one to a vertex. A guard on a vertex w can respond to an attack at a vertex v by moving along an edge from w to v (assuming v does not already have a guard). Informally, if such a response can be made, no matter which vertex is attacked changing position of the guards can continue to respond forever, we say that the guards form an eternally secure set. The 1-security problem and m-security problem are the two types of eternal security problems. In the 1-security problem, only one guard moves in response to an attack and in the m-security problem, more than one guard can move in response to an attack. Burger et al. [2,3] introduced 1-security problem and m-security problem are considered in [9,10,12]. The problem of two attacks with a single-guard movement is explored in [4,5]. The eternal 1-secure set of a graph G = (V, E) is defined as a set S0 ⊆ V that can defend against any sequence of single-vertex attacks by means of single-guard shifts along the edges of G. The 1-eternal security number is denoted by γ ∞ (G), as the minimum cardinality of an eternal 1-secure set. This parameter is introduced by Burger et al. [3] using the notation γ∞ . The eternal m-security number is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. A suitable placement of the guards is called an eternal m-secure set.
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The concept of an eternal total dominating set was introduced by Klostermeyer et al. [8], which further requires that each vertex with a guard to be adjacent to a vertex with a guard. The total dominating sets are a well-studied variant of dominating sets, hence it motivates to extend the notion of eternal domination to total domination. Klostermeyer et al. [8] defined an eternal dominating set (EDS) of G to be a set D such that for each sequence of attacks R = r1 , r2 , · · · with ri ∈ V there exists a sequence D = D1 , D2 , · · · of dominating sets and a sequence of vertices s1 , s2 , · · · where si ∈ Di ∩ N [ri ], such that Di+1 = (Di − {si })∪ {ri }. Note that si = ri is possible. The set Di+1 is the set of locations of guards after the attack at ri is defended. If si = ri , we say that the guard at si has moved to ri . The minimum cardinality amongst all eternal dominating sets is the eternal domination number γ ∞ (G). For the m-eternal dominating set problem, each Di , i ≥ 1, is required to be a dominating set, ri ∈ V (assuming without loss of generality ri ∈ Di ), and Di+1 is obtained from Di by moving the guards to neighbouring vertices. That is each guard in Di may move to an adjacent vertex, as long as one guard moves to ri . Thus it is required that ri ∈ Di+1 . The size of the smallest m-eternal dominating set (defined similar to an eternal dominating set) of G is the m-eternal domi∞ nation number γm (G). This ”multiple guards move” version of the problem was introduced by Goddard et al. [6]. It is also called the ”all guards move” ∞ model. It is clear that γ ∞ (G) ≥ γm (G) ≥ γ(G) for all graphs G. They defined an m-eternal total dominating set (m-ETDS) of G to be an EDS except that all the sets Di are total dominating sets. The minimum cardinal∞ ity amongst all m-ETDSs is the m-eternal total domination number γmt (G). ∞ A m-ETDS of minimum cardinality is called a γmt -set of G. The concept of graphs that are critical with respect to various operations on graphs have been studied by several authors. Brigham et al. [1] deals with the effect on the domination number of deleting vertices. These graphs are called γ-critical. For many graph parameters, criticality is a fundamental question. Many papers have been written about the graphs whose parameter (such as connectedness or chromatic number or domination number) goes up and down whenever an edge or vertex is removed or added. In this paper, ∞ the effects on γmt (G) when G is modified by deleting a vertex are examined. Further since deleting a vertex which is a support vertex from G will yield a ∞ graph with isolated vertices, we study the effects on γmt (G), when G is modified by deleting a non support vertex.
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2
Notation
We consider only finite simple undirected graph G = (V, E) with vertex set V (G) and edge set E(G). For graph theoretical terminology we refer to Harary [7]. For any vertex v ∈ V , the open neighbourhood of v is the set N (v) = {u ∈ V : uv ∈ E} and the closed neighbourhood is the setN [v] = N (v) ∪ v. For a set S ⊆ V , the open neighbourhood is N (S) = v∈S N (v) and the closed neighbourhood is N [S] = N (S)∪S. The external private neighbourhood epn(v,S) of a vertex v ∈ S is defined by epn(v, S) = {u ∈ V − S : N (u) ∩ S = {v}}. A vertex of degree one is called a pendant vertex. A vertex of G adjacent to pendant vertices is called a support. A support adjacent to exactly one pendant vertex is called weak support and a support adjacent to at least two pendant vertices is called a strong support. A connected graph having no cycles is called a tree. A graph G is called unicyclic, if it is connected and contains precisely one cycle. The Corona of graph G, is the graph obtained from G, by attaching a pendant vertex to each vertex of G. A Wheel graph Wn is a graph with n vertices, formed by connecting a single vertex to all vertices of an (n − 1) cycle.
3
Main Results
The changing and unchanging terminology was first suggested by Harary [7] and we examine these problems using this terminology. It is useful to partition the vertices of G into three sets according to how their removal ∞ affects γmt (G). Let G − v denote the graph formed by removing a vertex v from G, where v is not a support vertex and V = V 0 ∪ V + ∪ V − for ∞ ∞ V 0 = {v ∈ V : γmt (G − v) = γmt (G)} + ∞ ∞ V = {v ∈ V : γmt (G − v) > γmt (G)} ∞ ∞ V − = {v ∈ V : γmt (G − v) < γmt (G)} ∞ Consider a graph G and a γmt - set S of G. To defend an attack at a vertex w in V − S, a guard at some vertex x in N (w) ∩ S moves to w. There are occasions where a vertex in V − S can be defended by a unique vertex in S. In such cases we call the respective vertex in V − S as a sole private neighbour of the vertex in S which comes to its defence. ∞ Definition 3.1 [10] Let S be a γmt -set S of G and v ∈ S. A vertex w ∈ N (v)∩(V −S) is said to be a sole private neighbour of v, if v is the only member in S which can respond to an attack at w. The sole private neighbourhood spn(v, S) = {w ∈ N (v) ∩ (V − S) : w is a sole private neighbour of v } such that (S − {v}) ∪ {w} form an eternal total m- secure set of G.
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∞ Theorem 3.2 A vertex v ∈ V − if and only if, for some γmt -set S of G ∞ containing v, S ∩ V (Gi ) is a γmt -set of Gi where Gi , 1 ≤ i ≤ k, k ≥ 1, are the components of G − v. ∞ Proof. Let v ∈ V − and S be a γmt -set of G and G1 , G2 , . . . , Gk , k ≥ 1 be ∞ the components of G − v. We claim that Si = S ∩ V (Gi ) be a γmt -set of Gi , ∞ ∞ 1 ≤ i ≤ k. Suppose not. Then γmt (G) >| Si |. This implies γmt (G) =| Si | +1. ∞ Hence v ∈ V − , a contradiction. Hence Si is a γmt -set of Gi , for each i. ∞ Conversely suppose the condition holds for some γmt -set S of G containing − v. We claim that v ∈ V . Suppose not. Then clearly v ∈ V 0 and by the ∞ condition, S − {v} is a γmt -set of G, a contradiction to the minimality of S. − Hence v ∈ V . 2 ∞ Observation 1 If v ∈ V − , then there exists a γmt -set of G containing v such that | spn(v, S) |= 0. ∞ Theorem 3.3 For any graph G, v ∈ V + if and only if v is in every γmt -set of G. ∞ Proof. Let v ∈ V + . Suppose there exists a γmt -set of G such that v ∈ S. Then either v ∈ spn(u, s) for some u ∈ S or v is a non-private neighbour of S. ∞ ∞ In both the cases, we see that γmt (G − v) ≤ γmt (G), a contradiction. Hence v ∞ is in every γmt -set of G. ∞ Conversely, suppose v is in every γmt -set of G. If < spn(v, S) > does + not induce a clique, then clearly v ∈ V . If < spn(v, S) > induces a clique, then the guard at v also has to move for an attack at some vertex not in ∞ < spn(v, S) >. Otherwise, (S − {v}) ∪ {w}, w ∈< spn(v, S) > is also a γmt set of G. Hence there exists a sequence of vertices u1 , u2 , . . . , uk in S, where ∞ spn(u1 , S) = ∅, spn(uk , S) = ∅ and ui , 2 ≤ i ≤ k is in every γmt -set and v = ui , 2 ≤ i ≤ k, provided uk is not a support. Now, if there is an attack at some vertex x ∈ spn(ui , S), 2 ≤ i ≤ k, then the guards at u1 , u2 , . . . , ui have ∞ ∞ to move successively to respond to the attack. Hence γmt (G−v) ≥ γmt (G)+1. + Therefore v ∈ V . 2
Theorem 3.4 For any graph G, if x ∈ V + and y ∈ V − , then x and y are not adjacent. Proof. Suppose that x ∈ V + , y ∈ V − and xy ∈ E(G). By Observation ∞ 3.3, there exists a γmt -set S of G containing y such that spn(y, S) = ∅. By ∞ Theorem 3.4, x is in every γmt -set S of G. Therefore x ∈ S and |spn(x, S)| 1. Case(i) |spn(x, S)| = 1
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As discussed in the converse part of the proof of Theorem 3.4, we see that there exists a sequence of vertices u1 , u2 , . . . , uk in S, where spn(u1 , S) = ∅, spn(uk , S) = ∅ and < spn(uk , S) > does not induce a clique and ui , 2 ≤ i ≤ k, ∞ is in every γmt -set S of G and x = ui for some i, 2 ≤ i ≤ k. This implies that + ui ∈ V , 2 ≤ i ≤ k. Hence u1 = y and u2 = x. If there is an attack at w, w ∈ spn(uk , S) then the guards in the sequence u1 , u2 , . . . , uk where u1 = y and u2 = x move towards w and hence the guard at u1 = y move to the vertex x. Hence y ∈ V 0 . Case (ii) |spn(x, S)| ≥ 2. Clearly y ∈ V 0 . In both the cases, we see the y ∈ V 0 , a contradiction. Hence xy ∈ E(G). 2 We define a family of trees T as follows: A tree T ∈ T if the following conditions are satisfied. (i) T does not have a strong support. (ii) Each weak support is of degree two. (iii) Every non leaf vertex is either a support or adjacent to a support. (iv) Every non leaf vertex which is not a support is of degree at least three. Lemma 3.5 For any tree T ∈ T , V 0 = ∅. Proof. Let D = {V (T ) − L}, where L is set of all leaf vertices. Then D is a γt - set of T . If there is an attack at a leaf vertex w, then the guards in the (x2 , x1 )- path, where x1 , x2 are supports, w ∈ N (x1 ), move towards w leaving ∞ z, the leaf vertex in N (x2 ) undefended. Therefore γmt (T ) > γt (T ). Let S = {V (T )−L}∪{z} = D∪{z}, where z is a leaf vertex. Clearly S is a ∞ γmt - set of T . For, to defend an attack at any leaf vertex say x other than z, the guards in the (z, x)- path move one step towards x and now S = (S \ z) ∪ {x} is a T DS. Hence S can eternally respond to any sequence of attacks in G. ∞ Hence γmt (T ) = γt (T ) + 1. ∞ Now any non leaf non support vertex v is in every γmt - set of T . Therefore + ∞ ∞ by Theorem 3.4, v ∈ V . Further, for any leaf vertex v, γmt (T −v) = γmt (T )−1 − 0 which implies that v ∈ V . Therefore V = ∅. 2
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Theorem 3.6 For any tree T which is neither a corona graph nor a member ∞ ∞ of T , there exists a vertex u ∈ V (T ) such that γmt (T − u) = γmt (T ). ∞ Proof. Let S be a γmt -set of T . Suppose T contains a strong support. Let v be a strong support. Then clearly both v and a member of N (v) are in S. ∞ ∞ Hence γmt (T − u) = γmt (T ), where u is a leaf adjacent to v. Suppose T does not contain a strong support. In this case there exists a weak support v of T which is of degree two. Let u be the non leaf neighbour of v. We have the following cases.
Case (i) u is a weak support. There exists a non leaf neighbour of u, say w1 . If w1 is not a support, ∞ ∞ then γmt (T − w1 ) = γmt (T ). If w1 is a support and there exists a non leaf ∞ ∞ neighbour of w1 , say w2 which is not a support, then γmt (T −w2 ) = γmt (T ) . If w2 is a support, we repeat the process and in general there exists a vertex wk such that wk is not a support and w1 , w2 , . . . , wk−1 are all supports. Therefore ∞ ∞ γmt (T − wk ) = γmt (T ). Case (ii) u is not a weak support. Let z be the leaf neighbour of v. Clearly v ∈ S. If u, v, z ∈ S, then ∞ ∞ clearly spn(u, S) = ∅. If |spn(u, S)| = 1, then γmt (T − u) = γmt (T ). Here ∞ (S \ u) ∪ {w} is a γmt - set of T − u, where spn(u, S) = {w}. If |spn(u, S)| ≥ 2, ∞ ∞ ∞ then γmt (T −w) = γmt (T ), where w ∈ spn(u, S). Here S is a γmt - set of T −w. ∞ Otherwise, either u ∈ S, z ∈ / S or u ∈ / S, z ∈ S. In either case γmt (T −u) = ∞ ∞ γmt (T ). Further S or (S \ {u}) ∪ {z} is a γmt -set of T − u according as u ∈ /S or u ∈ S. 2 In [9] the following theorems have been proved. ∞ Theorem 3.7 [9] For Paths Pn , n ≥ 4, γmt (Pn ) = n2 + 1,
n > 1.
Theorem 3.8 [9] For cycles Cn , ∞ (Cn ) = γt (Cn ) = γmt
n 2
+ 1 , n ≡ 2(mod 4) otherwise.
n2 ,
Theorem 3.9 For Paths Pn , n ≥ 6, where n ≡ 0, 2(mod 4), V − = ∅. Proof. Let V (Pn ) = {v1 , v2 , . . . , vn }. Case (i) n ≡ 0(mod 4).
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∞ ∞ ∞ ∞ Now, γmt (Pn − v1 ) = γmt (Pn−1 ) = n−1 + 1 = γmt (Pn ), γmt (Pn − v3 ) = 2 ∞ ∞ ∞ ∞ ∞ ∞ γmt (P2 ) + γmt (Pn−3 ) > γmt (Pn ) and γmt (Pn − v4 ) = γmt (P3 ) + γmt (Pn−4 ) > ∞ 0 + γmt (Pn ). Hence v1 , v4 ∈ V , v3 ∈ V . ∞ ∞ ∞ For j ≥ 5, j = n − 1 or n, γmt (Pn − vj ) = γmt (Pk ) + γmt (Pn−k−1 ) where either k ≡ 0(mod 4) and n − k − 1 ≡ 3(mod 4) or vice versa or k ≡ 1(mod 4) and n − k − 1 ≡ 2(mod 4) or vice versa. Hence in both the cases vj ∈ V + .
Case (ii) n ≡ 2(mod 4). ∞ ∞ ∞ ∞ ∞ (Pn − v1 ) = γmt (Pn ), γmt (Pn − v3 ) = γmt (P2 ) + γmt (Pn−3 ) > Now, γmt ∞ ∞ ∞ ∞ ∞ γmt (Pn ) and γmt (Pn − v4 ) = γmt (P3 ) + γmt (Pn−4 ) = γmt (Pn ). Hence v3 ∈ V + , v 1 , v4 ∈ V 0 . ∞ ∞ ∞ For j ≥ 5, j = n − 1 or n , γmt (Pn − vj ) = γmt (Pk ) + γmt (Pn−k−1 ) where either k ≡ 0(mod 4) and n − k − 1 ≡ 1(mod 4) or vice versa or k ≡ 2(mod 4) and n − k − 1 ≡ 3(mod 4) or vice versa. Hence in both the cases vj ∈ V + . Therefore in all the above cases we see that vj ∈ V − , vj ∈ V 0 . 2 Theorem 3.10 For cycles Cn , n ≥ 3, where n ≡ 0(mod 4), V = V 0 . Proof. Let V (Cn ) = {v1 , v2 , . . . , vn }, n ≥ 3. If n ≡ 0(mod 4), then Cn − vi = ∞ ∞ Pn−1 , 1 ≤ i ≤ n, n − 1 ≡ 0, 1, 2(mod 4). Hence γmt (Cn − vi ) = γmt (Pn−1 ) = ∞ 0 γmt (Cn ). Therefore vi ∈ V , 1 ≤ i ≤ n. 2 Corollary 3.11 For cycles Cn , n ≥ 3, n ≡ 0(mod 4), V = V + . Proof. If vi , 1 ≤ i ≤ n, is removed from a cycle Cn , then the cycle becomes ∞ a path Pn−1 , where n − 1 ≡ 3(mod 4). Now, γmt (Cn ) = n2 < n2 + 1 = ∞ γmt (Pn−1 ). Hence V = V + . 2 Next, we explore a related problem of determining the minimum num∞+ ∞ (G). Let γmt (G) denote the miniber of vertices whose removal changes γmt ∞− ∞ mum number of vertices whose removal increases the γmt - value and γmt (G) ∞ the corresponding number whose removal decreases the γmt - value. Note that ∞− ∞+ γmt (G) may not exist. For example G = Kn . Note also that γmt (G) = 1 iff + V = ∅ . ∞+ ∞− In the following theorems, we find the values for γmt (G) + γmt (G) for paths and cycles. ∞+ ∞− Remark 3.12 For Paths Pn , n = 2, 3, neither γmt (Pn ) nor γmt (Pn ) exists. 1, n = 4,5,7 ∞+ ∞− (Pn )+γmt (Pn ) = Observation 2 For Paths 4 ≤ n ≤ 8, γmt 2, n = 6,8.
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∞+ ∞− Theorem 3.13 For n ≥ 9, γmt (Pn ) + γmt (Pn ) =
2, 3,
189
n ≡ 1, 3(mod 4) n ≡ 0, 2(mod 4).
∞ Proof. Let V (Pn ) = {v1 , v2 , . . . , vn }. By Theorem 3.8, γmt (Pn ) = n2 + 1.
Case (i) n ≡ 0, 2(mod 4). ∞ ∞ (Pn − vi ) = γmt (Pn ), where i = 1, n. The components of Pn − vi , Now γmt 3 ≤ i ≤ n − 2 are Pk and Pn−k−1 . If k is even, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−1 )
+
∞ γmt (Pn−k−1 )
n−k−1 k +1+ +1 = 2 2 n−k−1+1 k = +2+ 2 2 n = +2 2 ∞ > γmt (Pn ).
If k is odd, then ∞ γmt (Pk )
n−k−1 k +1+ +1 = 2 2 n−k−1 k+1 +2+ = 2 2 n = +2 2 ∞ > γmt (Pn ).
∞+ ∞− (Pn ) = 1 and γmt (Pn ) ≥ 2. Now, Therefore γmt ∞ ∞ (Pn − {v1 , v2 }) = γmt (Pn−2 ) γmt n−2 +1 = 2 n = 2 ∞ < γmt (Pn ). ∞− ∞+ ∞− (Pn ) = 2. Therefore γmt (Pn ) + γmt (Pn ) = 3. Hence γmt
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Case (ii) n ≡ 1, 3(mod 4). ∞ ∞ (Pn − v1 ) = γmt (Pn−1 ) γmt n−1 +1 = 2 n 1 = + 2n 2 = 2 ∞ (Pn ). < γmt ∞− (Pn ) = 1. Therefore γmt ∞ γmt (Pn
− v6 ) =
n−6 +1 =4+ 2 n 5 = + 2n 2 = +2 2 ∞ > γmt (Pn ).
∞ γmt (P5 )
+
∞ γmt (Pn−6 )
∞+ (Pn ) = 1. Further for all other components when i = 6, Hence γmt ∞+ ∞− ∞ ∞ γmt (Pn − vi ) = γmt (Pn ). Therefore γmt (Pn ) + γmt (Pn ) = 2. 2 2, n = 5,6 ∞+ ∞− . Theorem 3.14 γmt (Cn ) + γmt (Cn ) = 3, n=7 4, n ≡ 2(mod 4), n = 10 ∞+ ∞− Theorem 3.15 For n ≥ 8, γmt (Cn )+γmt (Cn ) = 5, n ≡ 2(mod 4).
Proof. Let V (Cn ) = {v1 , v2 , . . . , vn }. By Theorem 3.9, n + 1 , n ≡ 2(mod 4) ∞ (Cn ) = 2 n γt (Cn ) = γmt otherwise.
2 , ∞ ∞ (Cn − v) = γmt (Cn ) for all v ∈ Cn . Therefore When n ≡ 0(mod 4), γmt ∞+ ∞− γmt (Cn ) > 1 and γmt (Cn ) > 1.
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Case (i) n ≡ 0(mod 4). Let v be a vertex of Cn . Now, ∞ γmt (Cn
− v) =
∞ γmt (Pn−1 )
191
n−1 +1 = 2 n−1+1 = +1 2 n = +1 2 ∞ > γmt (Cn ).
∞+ (Cn ) = 1. Therefore γmt ∞− (Cn ). Let vi , vj be two vertices of Cn , such Now, we find the value of γmt that d(vi , vj ) = 2. If vi and vj are adjacent, then n−2 ∞ ∞ γmt (Cn − {vi , vj }) = γmt (Pn−2 ) = +1 2 n−2 = +1 2 n = 2 ∞ = γmt (Cn ).
If vi and vj are non-adjacent vertices, then the components of Cn − {vi , vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is even, then n−k−2 k ∞ ∞ +1+ +1 γmt (Pk ) + γmt (Pn−k−2 ) = 2 2 n−k−2 k = +2+ 2 2 n = +1 2 ∞ > γmt (Cn ). If k is odd, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−2 )
n−k−2 k +1+ +1 = 2 2 n−k−2+1 n k+1 +2+ = +2 = 2 2 2 ∞ > γmt (Cn ).
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∞ ∞ Hence in all the above cases, we see that γmt (Cn − {vi , vj }) > γmt (Cn ). Hence ∞− γmt (Cn ) ≥ 3. Now consider three vertices vi , vj , vl of Cn such that d(vi , vj ) = 2 and d(vj , vl ) = 2. If vi , vj , vl induce a P3 , then,
∞ ∞ (Cn − {vi , vj , vl }) = γmt (Pn−3 ) γmt n−3 +1 = 2 n−3+1 = +1 2 n = 2 ∞ = γmt (Cn ).
If vi , vj , vl induce a P2 ∪ K1 , then the components of Cn − {vi , vj , vl } are Pk and Pn−k−5 , k = 2, 3, . . . , n − 5 . If k is even, then ∞ ∞ ∞ γmt (Cn − {vi , vj , vl }) = γmt (Pk ) + γmt (Pn−k−5 ) n−k−5 k +1+ +1 = 2 2 n = +1 2 ∞ > γmt (Cn ).
If k is odd, then ∞ γmt (Pk )
+
∞ γmt (Pn−k−5 )
n−k−5 k +1+ +1 = 2 2 n = 2 ∞ = γmt (Cn ).
If vi , vj , vl induce a 3k1 , then the components of Cn − {vi , vj , vl } are Pk1 , Pk2 and Pk3 , where k1 + k2 + k3 = n − 3. Since n − 3 is odd, either k1 , k2 , k3 are all odd or two of the numbers k1 , k2 , k3 are even and one number is odd.
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If k1 , k2 , k3 are all odd, then k2 k3 k1 +1+ +1+ +1 = 2 2 2 k1 + 1 k2 + 1 k3 + 1 + + +3 = 2 2 2 n−3 9 + = 2 2 n = +3 2 ∞ > γmt (Cn ).
∞ γmt (Pk1 )
+
∞ γmt (Pk2 )
+
∞ γmt (Pk3 )
Without loss of generality let k1 , k2 be even and k3 be odd , then k2 k3 k1 +1+ +1+ +1 = 2 2 2 k1 k 2 k 3 + 1 + + +3 = 2 2 2 n−3 7 + = 2 2 n = +2 2 ∞ > γmt (Cn ).
∞ (Pk1 ) γmt
+
∞ γmt (Pk2 )
+
∞ γmt (Pk3 )
∞ ∞ Hence in all above cases, we see that γmt (Cn − {vi , vj , vl }) ≥ γmt (Cn ). There∞− fore γmt (Cn ) ≥ 4. Now
n−4 +1 = 2 n = −1 2 ∞ < γmt (Cn ).
∞ (Cn γmt
− {v1 , v2 , v3 , v4 }) =
∞ γmt (Pn−4 )
∞− ∞+ ∞− Hence γmt (Cn ) = 4. Therefore γmt (Cn ) + γmt (Cn ) = 5.
Case (ii) n ≡ 1(mod 4).
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Let vi , vj be two vertices of Cn such that d(vi , vj ) = 2. If vi , vj are adjacent, then n−2 ∞ ∞ γmt (Cn − {vi , vj }) = γmt (Pn−2 ) = +1 2 n−2+1 = +1 2 n 1 = + 2n 2 = 2 ∞ (Cn ). = γmt If vi and vj are non-adjacent vertices, then the components of Cn − {v, vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is odd, then n−k−2 k ∞ ∞ +1+ +1 γmt (Pk ) + γmt (Pn−k−2 ) = 2 2 n−k−2 k+1 +2+ = 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ). If k is even, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−2 )
n−k−2 k +1+ +1 = 2 2 n−k−2+1 k = +2+ 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ).
∞+ (Cn ) = 2. Hence γmt ∞− ∞− (Cn ), by the above argument we see that γmt (Cn ) ≥ 3. In order to find γmt Now,
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∞ ∞ γmt (Cn − {v1 , v2 , v3 }) = γmt (Pn−3 ) n−3 +1 = 2 n−3 = +1 2 n 1 = − 2n 2 = −1 2 ∞ < γmt (Cn ). ∞− ∞+ ∞− (Cn ) = 3. Therefore γmt (Cn ) + γmt (Cn ) = 5. Hence γmt
Case (iii) n ≡ 2(mod 4). ∞+ When n = 10, clearly one can verify that γmt (C10 ) does not exist. When n ≥ 14, n−2 +1 = 2 n = 2 ∞ < γmt (Cn ).
∞ γmt (Cn
− {v1 , v2 }) =
∞ γmt (Pn−2 )
∞− (Cn ) = 2. Therefore γmt Let vi and vj be non-adjacent vertices, then the components of Cn −{vi , vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is odd, then n−k−2 k ∞ ∞ +1+ +1 γmt (Pk ) + γmt (Pn−k−2 ) = 2 2 n−k−2+1 k+1 +2+ = 2 2 n = +2 2 ∞ > γmt (Cn ). ∞+ ∞+ ∞− (Cn ) = 2. Therefore γmt (Cn ) + γmt (Cn ) = 4. Here γmt
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Case (iv) n ≡ 3(mod 4). Let vi , vj be two vertices of Cn such that d(vi , vj ) = 2. If vi , vj are adjacent, then n−2 +1 = 2 n−2+1 = +1 2 n 1 = + 2n 2 = 2 ∞ = γmt (Cn ).
∞ γmt (Cn
− {vi , vj }) =
∞ γmt (Pn−2 )
If vi and vj are non-adjacent vertices, then the components of Cn − {vi , vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is even, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−2 )
n−k−2 k +1+ +1 = 2 2 n−k−2 k +1 = +1+ 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ).
If k is odd, then n−k−2 k+1 +1+ +1 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ).
∞ ∞ γmt (Pk ) + γmt (Pn−k−2 ) =
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∞+ ∞− Hence γmt (Cn ) = 2. By the above argument we see that γmt (Cn ) ≥ 3. Now
n−3 +1 = 2 n 1 = − 2n 2 = −1 2 ∞ < γmt (Cn ).
∞ γmt (Cn
− {v1 , v2 , v3 }) =
∞ γmt (Pn−3 )
∞− ∞+ ∞− Hence γmt (Cn ) = 3. Therefore γmt (Cn ) + γmt (Cn ) = 5.
2
∞+ ∞− (Wn ) nor γmt (Wn ) exists. Remark 3.16 For Wheels Wn , n < 6, neither γmt ∞+ ∞− (Wn ) + γmt (Wn ) = 1, n ≥ 6. Theorem 3.17 For Wheels Wn , γmt
Proof. Let V (Wn ) = {v, v1 , v2 , . . . , vn } where deg(v) = n−1 and deg(vi ) = 3, ∞− ∞ 1 ≤ i ≤ n − 1. Clearly γmt (Wn ) = 2. Hence γmt (Wn ) does not exist. Now ∞+ ∞+ we find the value of γmt (Wn ). For n < 6, γmt (Wn ) does not exist. When ∞ n = 6, removal of v from Wn , results in a cycle Cn and γmt (C5 ) = 3. When ∞ n ≥ 7, removal of v from Wn , results in a cycle Cn−1 and γmt (Cn−1 ) ≥ 3. ∞+ ∞+ ∞− Hence γmt (Wn ) = 1. Hence γmt (Wn ) + γmt (Wn ) = 1. 2
References [1] Brigham R.C., P. Chinn and R.D.Dutton, Vertex domination critical graphs, Networks, 18 (1988), 173–179. [2] Burger A. P., E. J. Cockayne, W. R. Grundlingh, C. M. Mynhardt, J. H. Van Vuuren and W. Winterbach, Finite order domination in graphs, J. Combin. Math. Combin. Comput., 49 (2004), 159–175. [3] Burger A. P., E. J. Cockayne, W. R. Grundlingh, C. M. Mynhardt, J. H. Van Vuuren and W. Winterbach, Infinite order domination in graphs, J. Combin. Math. Combin. Comput., 52 (2005), 169–180. [4] Cockayne E. J., O. Favaron and C. M. Mynhardt, Secure domination, weak Roman domination and forbidden subgraphs, Bull. Inst. Combin. Appl., 39 (2003), 87–100. [5] Cockayne E. J., P. J. P. Grobler, W. R. Grundlingh, J. Munganga and J. H. Van Vuuren, Protection of a graph, Util. Math., 67 (2005), 19–32.
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[6] Goddard W., S. M. Hedetniemi and S. T. Hedetniemi, Eternal security in graphs, J. Combin. Comput., 52 (2005), 169–180. [7] Harary F., Graph theory, Addison Wesley, Reading, Mass, 1969. [8] Klostermeyer W.F. and C. M. Mynhardt, Eternal Total Domination in Graphs, Ars Combin., 107 (2012), 473–492. [9] Roushini Leely Pushpam P. and PA.Shanthi, m-Eternal Total Domination in graphs, Submitted for Publication. [10] Roushini Leely Pushpam P. and G. Navamani, Changing and Unchanging in Eternal m-Security in Graphs, Submitted for Publication. [11] Roushini Leely Pushpam P. and G. Navamani, Eternal m-Security in Certain classes of Graphs, J. Combin. Math. Combin. Comput., 92 (2015), 25-38. [12] Roushini Leely Pushpam P. and G. Navamani, Eternal m-Security in Graphs, International Journal of Mathematics and Soft Computing, 5(2) (2015), 115–125.
Electronic Notes in Discrete Mathematics 53 (2016) 181–198 www.elsevier.com/locate/endm
Changing and Unchanging in m-Eternal Total Domination in Graphs P. Roushini Leely Pushpam1 Department of Mathematics D.B. Jain College Chennai 600 097, Tamil Nadu, India
PA. Shanthi2 Department of Mathematics Sri Sairam Engineering College Chennai 600 044, Tamil Nadu, India Abstract The Eternal dominating set of a graph is defined as a set of guards located at vertices, required to protect the vertices of the graph against infinitely long sequences of attacks, such that the configuration of guards induces a dominating set at all times. The eternal m-security number is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. Klostermeyer and Mynhardt introduced the concept of an m-eternal total dominating set, which further requires that each vertex with a guard to be adjacent to a vertex with a guard. They defined the m-eternal total domination number of a graph G denoted ∞ (G) as the minimum number of guards to handle an arbitrary sequence of by γmt single attacks using multiple guard shifts and the configuration of guards always ∞ (G) induces a total dominating set. In this paper we examine the effects on γmt when G is modified by deleting a vertex which is a non support vertex. Keywords: Eternal total domination, total domination. 1 2
Email: e-mail: [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2016.05.017 1571-0653/© 2016 Elsevier B.V. All rights reserved.
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1
Introduction
Let G = (V, E) be a simple and connected graph of order |V | = n. For graph theoretic terminology, we refer to Harary [7]. For any vertex v ∈ V , the open neighbourhood of v is the set N (v) = {u ∈ V |uv ∈ E} and the closed neighbourhood is the set N [v] = N (v) ∪ {v}. For a set S ⊆ V , the open neighbourhood is N (S) = ∪v∈s N (v) and the closed neighbourhood is N [S] = N (S) ∪ S. A set S is a dominating set, if N [S] = V (G) or equivalently, every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G, and a dominating set S of minimum cardinality is called a γ-set of G. A total dominating set (TDS) of G is a set D ⊆ V with the property that for each u ∈ V , there exists x ∈ D adjacent to u. The minimum cardinality amongst all total dominating sets is the total domination number γt (G). A TDS S of minimum cardinality is called γt -set of G. It can be noted that this parameter is only defined for graphs without isolated vertices. Burger et al. [2,3] introduced a dynamic form of domination which has been designated eternal security by Goddard et al. [6]. In this concept, fixed number of guards are positioned on the vertices of a graph G = (V, E), at most one to a vertex. A guard on a vertex w can respond to an attack at a vertex v by moving along an edge from w to v (assuming v does not already have a guard). Informally, if such a response can be made, no matter which vertex is attacked changing position of the guards can continue to respond forever, we say that the guards form an eternally secure set. The 1-security problem and m-security problem are the two types of eternal security problems. In the 1-security problem, only one guard moves in response to an attack and in the m-security problem, more than one guard can move in response to an attack. Burger et al. [2,3] introduced 1-security problem and m-security problem are considered in [9,10,12]. The problem of two attacks with a single-guard movement is explored in [4,5]. The eternal 1-secure set of a graph G = (V, E) is defined as a set S0 ⊆ V that can defend against any sequence of single-vertex attacks by means of single-guard shifts along the edges of G. The 1-eternal security number is denoted by γ ∞ (G), as the minimum cardinality of an eternal 1-secure set. This parameter is introduced by Burger et al. [3] using the notation γ∞ . The eternal m-security number is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. A suitable placement of the guards is called an eternal m-secure set.
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The concept of an eternal total dominating set was introduced by Klostermeyer et al. [8], which further requires that each vertex with a guard to be adjacent to a vertex with a guard. The total dominating sets are a well-studied variant of dominating sets, hence it motivates to extend the notion of eternal domination to total domination. Klostermeyer et al. [8] defined an eternal dominating set (EDS) of G to be a set D such that for each sequence of attacks R = r1 , r2 , · · · with ri ∈ V there exists a sequence D = D1 , D2 , · · · of dominating sets and a sequence of vertices s1 , s2 , · · · where si ∈ Di ∩ N [ri ], such that Di+1 = (Di − {si })∪ {ri }. Note that si = ri is possible. The set Di+1 is the set of locations of guards after the attack at ri is defended. If si = ri , we say that the guard at si has moved to ri . The minimum cardinality amongst all eternal dominating sets is the eternal domination number γ ∞ (G). For the m-eternal dominating set problem, each Di , i ≥ 1, is required to be a dominating set, ri ∈ V (assuming without loss of generality ri ∈ Di ), and Di+1 is obtained from Di by moving the guards to neighbouring vertices. That is each guard in Di may move to an adjacent vertex, as long as one guard moves to ri . Thus it is required that ri ∈ Di+1 . The size of the smallest m-eternal dominating set (defined similar to an eternal dominating set) of G is the m-eternal domi∞ nation number γm (G). This ”multiple guards move” version of the problem was introduced by Goddard et al. [6]. It is also called the ”all guards move” ∞ model. It is clear that γ ∞ (G) ≥ γm (G) ≥ γ(G) for all graphs G. They defined an m-eternal total dominating set (m-ETDS) of G to be an EDS except that all the sets Di are total dominating sets. The minimum cardinal∞ ity amongst all m-ETDSs is the m-eternal total domination number γmt (G). ∞ A m-ETDS of minimum cardinality is called a γmt -set of G. The concept of graphs that are critical with respect to various operations on graphs have been studied by several authors. Brigham et al. [1] deals with the effect on the domination number of deleting vertices. These graphs are called γ-critical. For many graph parameters, criticality is a fundamental question. Many papers have been written about the graphs whose parameter (such as connectedness or chromatic number or domination number) goes up and down whenever an edge or vertex is removed or added. In this paper, ∞ the effects on γmt (G) when G is modified by deleting a vertex are examined. Further since deleting a vertex which is a support vertex from G will yield a ∞ graph with isolated vertices, we study the effects on γmt (G), when G is modified by deleting a non support vertex.
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2
Notation
We consider only finite simple undirected graph G = (V, E) with vertex set V (G) and edge set E(G). For graph theoretical terminology we refer to Harary [7]. For any vertex v ∈ V , the open neighbourhood of v is the set N (v) = {u ∈ V : uv ∈ E} and the closed neighbourhood is the setN [v] = N (v) ∪ v. For a set S ⊆ V , the open neighbourhood is N (S) = v∈S N (v) and the closed neighbourhood is N [S] = N (S)∪S. The external private neighbourhood epn(v,S) of a vertex v ∈ S is defined by epn(v, S) = {u ∈ V − S : N (u) ∩ S = {v}}. A vertex of degree one is called a pendant vertex. A vertex of G adjacent to pendant vertices is called a support. A support adjacent to exactly one pendant vertex is called weak support and a support adjacent to at least two pendant vertices is called a strong support. A connected graph having no cycles is called a tree. A graph G is called unicyclic, if it is connected and contains precisely one cycle. The Corona of graph G, is the graph obtained from G, by attaching a pendant vertex to each vertex of G. A Wheel graph Wn is a graph with n vertices, formed by connecting a single vertex to all vertices of an (n − 1) cycle.
3
Main Results
The changing and unchanging terminology was first suggested by Harary [7] and we examine these problems using this terminology. It is useful to partition the vertices of G into three sets according to how their removal ∞ affects γmt (G). Let G − v denote the graph formed by removing a vertex v from G, where v is not a support vertex and V = V 0 ∪ V + ∪ V − for ∞ ∞ V 0 = {v ∈ V : γmt (G − v) = γmt (G)} + ∞ ∞ V = {v ∈ V : γmt (G − v) > γmt (G)} ∞ ∞ V − = {v ∈ V : γmt (G − v) < γmt (G)} ∞ Consider a graph G and a γmt - set S of G. To defend an attack at a vertex w in V − S, a guard at some vertex x in N (w) ∩ S moves to w. There are occasions where a vertex in V − S can be defended by a unique vertex in S. In such cases we call the respective vertex in V − S as a sole private neighbour of the vertex in S which comes to its defence. ∞ Definition 3.1 [10] Let S be a γmt -set S of G and v ∈ S. A vertex w ∈ N (v)∩(V −S) is said to be a sole private neighbour of v, if v is the only member in S which can respond to an attack at w. The sole private neighbourhood spn(v, S) = {w ∈ N (v) ∩ (V − S) : w is a sole private neighbour of v } such that (S − {v}) ∪ {w} form an eternal total m- secure set of G.
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∞ Theorem 3.2 A vertex v ∈ V − if and only if, for some γmt -set S of G ∞ containing v, S ∩ V (Gi ) is a γmt -set of Gi where Gi , 1 ≤ i ≤ k, k ≥ 1, are the components of G − v. ∞ Proof. Let v ∈ V − and S be a γmt -set of G and G1 , G2 , . . . , Gk , k ≥ 1 be ∞ the components of G − v. We claim that Si = S ∩ V (Gi ) be a γmt -set of Gi , ∞ ∞ 1 ≤ i ≤ k. Suppose not. Then γmt (G) >| Si |. This implies γmt (G) =| Si | +1. ∞ Hence v ∈ V − , a contradiction. Hence Si is a γmt -set of Gi , for each i. ∞ Conversely suppose the condition holds for some γmt -set S of G containing − v. We claim that v ∈ V . Suppose not. Then clearly v ∈ V 0 and by the ∞ condition, S − {v} is a γmt -set of G, a contradiction to the minimality of S. − Hence v ∈ V . 2 ∞ Observation 1 If v ∈ V − , then there exists a γmt -set of G containing v such that | spn(v, S) |= 0. ∞ Theorem 3.3 For any graph G, v ∈ V + if and only if v is in every γmt -set of G. ∞ Proof. Let v ∈ V + . Suppose there exists a γmt -set of G such that v ∈ S. Then either v ∈ spn(u, s) for some u ∈ S or v is a non-private neighbour of S. ∞ ∞ In both the cases, we see that γmt (G − v) ≤ γmt (G), a contradiction. Hence v ∞ is in every γmt -set of G. ∞ Conversely, suppose v is in every γmt -set of G. If < spn(v, S) > does + not induce a clique, then clearly v ∈ V . If < spn(v, S) > induces a clique, then the guard at v also has to move for an attack at some vertex not in ∞ < spn(v, S) >. Otherwise, (S − {v}) ∪ {w}, w ∈< spn(v, S) > is also a γmt set of G. Hence there exists a sequence of vertices u1 , u2 , . . . , uk in S, where ∞ spn(u1 , S) = ∅, spn(uk , S) = ∅ and ui , 2 ≤ i ≤ k is in every γmt -set and v = ui , 2 ≤ i ≤ k, provided uk is not a support. Now, if there is an attack at some vertex x ∈ spn(ui , S), 2 ≤ i ≤ k, then the guards at u1 , u2 , . . . , ui have ∞ ∞ to move successively to respond to the attack. Hence γmt (G−v) ≥ γmt (G)+1. + Therefore v ∈ V . 2
Theorem 3.4 For any graph G, if x ∈ V + and y ∈ V − , then x and y are not adjacent. Proof. Suppose that x ∈ V + , y ∈ V − and xy ∈ E(G). By Observation ∞ 3.3, there exists a γmt -set S of G containing y such that spn(y, S) = ∅. By ∞ Theorem 3.4, x is in every γmt -set S of G. Therefore x ∈ S and |spn(x, S)| 1. Case(i) |spn(x, S)| = 1
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As discussed in the converse part of the proof of Theorem 3.4, we see that there exists a sequence of vertices u1 , u2 , . . . , uk in S, where spn(u1 , S) = ∅, spn(uk , S) = ∅ and < spn(uk , S) > does not induce a clique and ui , 2 ≤ i ≤ k, ∞ is in every γmt -set S of G and x = ui for some i, 2 ≤ i ≤ k. This implies that + ui ∈ V , 2 ≤ i ≤ k. Hence u1 = y and u2 = x. If there is an attack at w, w ∈ spn(uk , S) then the guards in the sequence u1 , u2 , . . . , uk where u1 = y and u2 = x move towards w and hence the guard at u1 = y move to the vertex x. Hence y ∈ V 0 . Case (ii) |spn(x, S)| ≥ 2. Clearly y ∈ V 0 . In both the cases, we see the y ∈ V 0 , a contradiction. Hence xy ∈ E(G). 2 We define a family of trees T as follows: A tree T ∈ T if the following conditions are satisfied. (i) T does not have a strong support. (ii) Each weak support is of degree two. (iii) Every non leaf vertex is either a support or adjacent to a support. (iv) Every non leaf vertex which is not a support is of degree at least three. Lemma 3.5 For any tree T ∈ T , V 0 = ∅. Proof. Let D = {V (T ) − L}, where L is set of all leaf vertices. Then D is a γt - set of T . If there is an attack at a leaf vertex w, then the guards in the (x2 , x1 )- path, where x1 , x2 are supports, w ∈ N (x1 ), move towards w leaving ∞ z, the leaf vertex in N (x2 ) undefended. Therefore γmt (T ) > γt (T ). Let S = {V (T )−L}∪{z} = D∪{z}, where z is a leaf vertex. Clearly S is a ∞ γmt - set of T . For, to defend an attack at any leaf vertex say x other than z, the guards in the (z, x)- path move one step towards x and now S = (S \ z) ∪ {x} is a T DS. Hence S can eternally respond to any sequence of attacks in G. ∞ Hence γmt (T ) = γt (T ) + 1. ∞ Now any non leaf non support vertex v is in every γmt - set of T . Therefore + ∞ ∞ by Theorem 3.4, v ∈ V . Further, for any leaf vertex v, γmt (T −v) = γmt (T )−1 − 0 which implies that v ∈ V . Therefore V = ∅. 2
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Theorem 3.6 For any tree T which is neither a corona graph nor a member ∞ ∞ of T , there exists a vertex u ∈ V (T ) such that γmt (T − u) = γmt (T ). ∞ Proof. Let S be a γmt -set of T . Suppose T contains a strong support. Let v be a strong support. Then clearly both v and a member of N (v) are in S. ∞ ∞ Hence γmt (T − u) = γmt (T ), where u is a leaf adjacent to v. Suppose T does not contain a strong support. In this case there exists a weak support v of T which is of degree two. Let u be the non leaf neighbour of v. We have the following cases.
Case (i) u is a weak support. There exists a non leaf neighbour of u, say w1 . If w1 is not a support, ∞ ∞ then γmt (T − w1 ) = γmt (T ). If w1 is a support and there exists a non leaf ∞ ∞ neighbour of w1 , say w2 which is not a support, then γmt (T −w2 ) = γmt (T ) . If w2 is a support, we repeat the process and in general there exists a vertex wk such that wk is not a support and w1 , w2 , . . . , wk−1 are all supports. Therefore ∞ ∞ γmt (T − wk ) = γmt (T ). Case (ii) u is not a weak support. Let z be the leaf neighbour of v. Clearly v ∈ S. If u, v, z ∈ S, then ∞ ∞ clearly spn(u, S) = ∅. If |spn(u, S)| = 1, then γmt (T − u) = γmt (T ). Here ∞ (S \ u) ∪ {w} is a γmt - set of T − u, where spn(u, S) = {w}. If |spn(u, S)| ≥ 2, ∞ ∞ ∞ then γmt (T −w) = γmt (T ), where w ∈ spn(u, S). Here S is a γmt - set of T −w. ∞ Otherwise, either u ∈ S, z ∈ / S or u ∈ / S, z ∈ S. In either case γmt (T −u) = ∞ ∞ γmt (T ). Further S or (S \ {u}) ∪ {z} is a γmt -set of T − u according as u ∈ /S or u ∈ S. 2 In [9] the following theorems have been proved. ∞ Theorem 3.7 [9] For Paths Pn , n ≥ 4, γmt (Pn ) = n2 + 1,
n > 1.
Theorem 3.8 [9] For cycles Cn , ∞ (Cn ) = γt (Cn ) = γmt
n 2
+ 1 , n ≡ 2(mod 4) otherwise.
n2 ,
Theorem 3.9 For Paths Pn , n ≥ 6, where n ≡ 0, 2(mod 4), V − = ∅. Proof. Let V (Pn ) = {v1 , v2 , . . . , vn }. Case (i) n ≡ 0(mod 4).
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∞ ∞ ∞ ∞ Now, γmt (Pn − v1 ) = γmt (Pn−1 ) = n−1 + 1 = γmt (Pn ), γmt (Pn − v3 ) = 2 ∞ ∞ ∞ ∞ ∞ ∞ γmt (P2 ) + γmt (Pn−3 ) > γmt (Pn ) and γmt (Pn − v4 ) = γmt (P3 ) + γmt (Pn−4 ) > ∞ 0 + γmt (Pn ). Hence v1 , v4 ∈ V , v3 ∈ V . ∞ ∞ ∞ For j ≥ 5, j = n − 1 or n, γmt (Pn − vj ) = γmt (Pk ) + γmt (Pn−k−1 ) where either k ≡ 0(mod 4) and n − k − 1 ≡ 3(mod 4) or vice versa or k ≡ 1(mod 4) and n − k − 1 ≡ 2(mod 4) or vice versa. Hence in both the cases vj ∈ V + .
Case (ii) n ≡ 2(mod 4). ∞ ∞ ∞ ∞ ∞ (Pn − v1 ) = γmt (Pn ), γmt (Pn − v3 ) = γmt (P2 ) + γmt (Pn−3 ) > Now, γmt ∞ ∞ ∞ ∞ ∞ γmt (Pn ) and γmt (Pn − v4 ) = γmt (P3 ) + γmt (Pn−4 ) = γmt (Pn ). Hence v3 ∈ V + , v 1 , v4 ∈ V 0 . ∞ ∞ ∞ For j ≥ 5, j = n − 1 or n , γmt (Pn − vj ) = γmt (Pk ) + γmt (Pn−k−1 ) where either k ≡ 0(mod 4) and n − k − 1 ≡ 1(mod 4) or vice versa or k ≡ 2(mod 4) and n − k − 1 ≡ 3(mod 4) or vice versa. Hence in both the cases vj ∈ V + . Therefore in all the above cases we see that vj ∈ V − , vj ∈ V 0 . 2 Theorem 3.10 For cycles Cn , n ≥ 3, where n ≡ 0(mod 4), V = V 0 . Proof. Let V (Cn ) = {v1 , v2 , . . . , vn }, n ≥ 3. If n ≡ 0(mod 4), then Cn − vi = ∞ ∞ Pn−1 , 1 ≤ i ≤ n, n − 1 ≡ 0, 1, 2(mod 4). Hence γmt (Cn − vi ) = γmt (Pn−1 ) = ∞ 0 γmt (Cn ). Therefore vi ∈ V , 1 ≤ i ≤ n. 2 Corollary 3.11 For cycles Cn , n ≥ 3, n ≡ 0(mod 4), V = V + . Proof. If vi , 1 ≤ i ≤ n, is removed from a cycle Cn , then the cycle becomes ∞ a path Pn−1 , where n − 1 ≡ 3(mod 4). Now, γmt (Cn ) = n2 < n2 + 1 = ∞ γmt (Pn−1 ). Hence V = V + . 2 Next, we explore a related problem of determining the minimum num∞+ ∞ (G). Let γmt (G) denote the miniber of vertices whose removal changes γmt ∞− ∞ mum number of vertices whose removal increases the γmt - value and γmt (G) ∞ the corresponding number whose removal decreases the γmt - value. Note that ∞− ∞+ γmt (G) may not exist. For example G = Kn . Note also that γmt (G) = 1 iff + V = ∅ . ∞+ ∞− In the following theorems, we find the values for γmt (G) + γmt (G) for paths and cycles. ∞+ ∞− Remark 3.12 For Paths Pn , n = 2, 3, neither γmt (Pn ) nor γmt (Pn ) exists. 1, n = 4,5,7 ∞+ ∞− (Pn )+γmt (Pn ) = Observation 2 For Paths 4 ≤ n ≤ 8, γmt 2, n = 6,8.
P. Roushini Leely Pushpam, P.A. Shanthi / Electronic Notes in Discrete Mathematics 53 (2016) 181–198
∞+ ∞− Theorem 3.13 For n ≥ 9, γmt (Pn ) + γmt (Pn ) =
2, 3,
189
n ≡ 1, 3(mod 4) n ≡ 0, 2(mod 4).
∞ Proof. Let V (Pn ) = {v1 , v2 , . . . , vn }. By Theorem 3.8, γmt (Pn ) = n2 + 1.
Case (i) n ≡ 0, 2(mod 4). ∞ ∞ (Pn − vi ) = γmt (Pn ), where i = 1, n. The components of Pn − vi , Now γmt 3 ≤ i ≤ n − 2 are Pk and Pn−k−1 . If k is even, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−1 )
+
∞ γmt (Pn−k−1 )
n−k−1 k +1+ +1 = 2 2 n−k−1+1 k = +2+ 2 2 n = +2 2 ∞ > γmt (Pn ).
If k is odd, then ∞ γmt (Pk )
n−k−1 k +1+ +1 = 2 2 n−k−1 k+1 +2+ = 2 2 n = +2 2 ∞ > γmt (Pn ).
∞+ ∞− (Pn ) = 1 and γmt (Pn ) ≥ 2. Now, Therefore γmt ∞ ∞ (Pn − {v1 , v2 }) = γmt (Pn−2 ) γmt n−2 +1 = 2 n = 2 ∞ < γmt (Pn ). ∞− ∞+ ∞− (Pn ) = 2. Therefore γmt (Pn ) + γmt (Pn ) = 3. Hence γmt
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Case (ii) n ≡ 1, 3(mod 4). ∞ ∞ (Pn − v1 ) = γmt (Pn−1 ) γmt n−1 +1 = 2 n 1 = + 2n 2 = 2 ∞ (Pn ). < γmt ∞− (Pn ) = 1. Therefore γmt ∞ γmt (Pn
− v6 ) =
n−6 +1 =4+ 2 n 5 = + 2n 2 = +2 2 ∞ > γmt (Pn ).
∞ γmt (P5 )
+
∞ γmt (Pn−6 )
∞+ (Pn ) = 1. Further for all other components when i = 6, Hence γmt ∞+ ∞− ∞ ∞ γmt (Pn − vi ) = γmt (Pn ). Therefore γmt (Pn ) + γmt (Pn ) = 2. 2 2, n = 5,6 ∞+ ∞− . Theorem 3.14 γmt (Cn ) + γmt (Cn ) = 3, n=7 4, n ≡ 2(mod 4), n = 10 ∞+ ∞− Theorem 3.15 For n ≥ 8, γmt (Cn )+γmt (Cn ) = 5, n ≡ 2(mod 4).
Proof. Let V (Cn ) = {v1 , v2 , . . . , vn }. By Theorem 3.9, n + 1 , n ≡ 2(mod 4) ∞ (Cn ) = 2 n γt (Cn ) = γmt otherwise.
2 , ∞ ∞ (Cn − v) = γmt (Cn ) for all v ∈ Cn . Therefore When n ≡ 0(mod 4), γmt ∞+ ∞− γmt (Cn ) > 1 and γmt (Cn ) > 1.
P. Roushini Leely Pushpam, P.A. Shanthi / Electronic Notes in Discrete Mathematics 53 (2016) 181–198
Case (i) n ≡ 0(mod 4). Let v be a vertex of Cn . Now, ∞ γmt (Cn
− v) =
∞ γmt (Pn−1 )
191
n−1 +1 = 2 n−1+1 = +1 2 n = +1 2 ∞ > γmt (Cn ).
∞+ (Cn ) = 1. Therefore γmt ∞− (Cn ). Let vi , vj be two vertices of Cn , such Now, we find the value of γmt that d(vi , vj ) = 2. If vi and vj are adjacent, then n−2 ∞ ∞ γmt (Cn − {vi , vj }) = γmt (Pn−2 ) = +1 2 n−2 = +1 2 n = 2 ∞ = γmt (Cn ).
If vi and vj are non-adjacent vertices, then the components of Cn − {vi , vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is even, then n−k−2 k ∞ ∞ +1+ +1 γmt (Pk ) + γmt (Pn−k−2 ) = 2 2 n−k−2 k = +2+ 2 2 n = +1 2 ∞ > γmt (Cn ). If k is odd, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−2 )
n−k−2 k +1+ +1 = 2 2 n−k−2+1 n k+1 +2+ = +2 = 2 2 2 ∞ > γmt (Cn ).
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∞ ∞ Hence in all the above cases, we see that γmt (Cn − {vi , vj }) > γmt (Cn ). Hence ∞− γmt (Cn ) ≥ 3. Now consider three vertices vi , vj , vl of Cn such that d(vi , vj ) = 2 and d(vj , vl ) = 2. If vi , vj , vl induce a P3 , then,
∞ ∞ (Cn − {vi , vj , vl }) = γmt (Pn−3 ) γmt n−3 +1 = 2 n−3+1 = +1 2 n = 2 ∞ = γmt (Cn ).
If vi , vj , vl induce a P2 ∪ K1 , then the components of Cn − {vi , vj , vl } are Pk and Pn−k−5 , k = 2, 3, . . . , n − 5 . If k is even, then ∞ ∞ ∞ γmt (Cn − {vi , vj , vl }) = γmt (Pk ) + γmt (Pn−k−5 ) n−k−5 k +1+ +1 = 2 2 n = +1 2 ∞ > γmt (Cn ).
If k is odd, then ∞ γmt (Pk )
+
∞ γmt (Pn−k−5 )
n−k−5 k +1+ +1 = 2 2 n = 2 ∞ = γmt (Cn ).
If vi , vj , vl induce a 3k1 , then the components of Cn − {vi , vj , vl } are Pk1 , Pk2 and Pk3 , where k1 + k2 + k3 = n − 3. Since n − 3 is odd, either k1 , k2 , k3 are all odd or two of the numbers k1 , k2 , k3 are even and one number is odd.
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If k1 , k2 , k3 are all odd, then k2 k3 k1 +1+ +1+ +1 = 2 2 2 k1 + 1 k2 + 1 k3 + 1 + + +3 = 2 2 2 n−3 9 + = 2 2 n = +3 2 ∞ > γmt (Cn ).
∞ γmt (Pk1 )
+
∞ γmt (Pk2 )
+
∞ γmt (Pk3 )
Without loss of generality let k1 , k2 be even and k3 be odd , then k2 k3 k1 +1+ +1+ +1 = 2 2 2 k1 k 2 k 3 + 1 + + +3 = 2 2 2 n−3 7 + = 2 2 n = +2 2 ∞ > γmt (Cn ).
∞ (Pk1 ) γmt
+
∞ γmt (Pk2 )
+
∞ γmt (Pk3 )
∞ ∞ Hence in all above cases, we see that γmt (Cn − {vi , vj , vl }) ≥ γmt (Cn ). There∞− fore γmt (Cn ) ≥ 4. Now
n−4 +1 = 2 n = −1 2 ∞ < γmt (Cn ).
∞ (Cn γmt
− {v1 , v2 , v3 , v4 }) =
∞ γmt (Pn−4 )
∞− ∞+ ∞− Hence γmt (Cn ) = 4. Therefore γmt (Cn ) + γmt (Cn ) = 5.
Case (ii) n ≡ 1(mod 4).
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Let vi , vj be two vertices of Cn such that d(vi , vj ) = 2. If vi , vj are adjacent, then n−2 ∞ ∞ γmt (Cn − {vi , vj }) = γmt (Pn−2 ) = +1 2 n−2+1 = +1 2 n 1 = + 2n 2 = 2 ∞ (Cn ). = γmt If vi and vj are non-adjacent vertices, then the components of Cn − {v, vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is odd, then n−k−2 k ∞ ∞ +1+ +1 γmt (Pk ) + γmt (Pn−k−2 ) = 2 2 n−k−2 k+1 +2+ = 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ). If k is even, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−2 )
n−k−2 k +1+ +1 = 2 2 n−k−2+1 k = +2+ 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ).
∞+ (Cn ) = 2. Hence γmt ∞− ∞− (Cn ), by the above argument we see that γmt (Cn ) ≥ 3. In order to find γmt Now,
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195
∞ ∞ γmt (Cn − {v1 , v2 , v3 }) = γmt (Pn−3 ) n−3 +1 = 2 n−3 = +1 2 n 1 = − 2n 2 = −1 2 ∞ < γmt (Cn ). ∞− ∞+ ∞− (Cn ) = 3. Therefore γmt (Cn ) + γmt (Cn ) = 5. Hence γmt
Case (iii) n ≡ 2(mod 4). ∞+ When n = 10, clearly one can verify that γmt (C10 ) does not exist. When n ≥ 14, n−2 +1 = 2 n = 2 ∞ < γmt (Cn ).
∞ γmt (Cn
− {v1 , v2 }) =
∞ γmt (Pn−2 )
∞− (Cn ) = 2. Therefore γmt Let vi and vj be non-adjacent vertices, then the components of Cn −{vi , vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is odd, then n−k−2 k ∞ ∞ +1+ +1 γmt (Pk ) + γmt (Pn−k−2 ) = 2 2 n−k−2+1 k+1 +2+ = 2 2 n = +2 2 ∞ > γmt (Cn ). ∞+ ∞+ ∞− (Cn ) = 2. Therefore γmt (Cn ) + γmt (Cn ) = 4. Here γmt
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Case (iv) n ≡ 3(mod 4). Let vi , vj be two vertices of Cn such that d(vi , vj ) = 2. If vi , vj are adjacent, then n−2 +1 = 2 n−2+1 = +1 2 n 1 = + 2n 2 = 2 ∞ = γmt (Cn ).
∞ γmt (Cn
− {vi , vj }) =
∞ γmt (Pn−2 )
If vi and vj are non-adjacent vertices, then the components of Cn − {vi , vj } are Pk and Pn−k−2 , k = 2, 3, . . . , n − 4. If k is even, then ∞ (Pk ) γmt
+
∞ γmt (Pn−k−2 )
n−k−2 k +1+ +1 = 2 2 n−k−2 k +1 = +1+ 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ).
If k is odd, then n−k−2 k+1 +1+ +1 2 2 n 3 = + 2n 2 = +1 2 ∞ > γmt (Cn ).
∞ ∞ γmt (Pk ) + γmt (Pn−k−2 ) =
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197
∞+ ∞− Hence γmt (Cn ) = 2. By the above argument we see that γmt (Cn ) ≥ 3. Now
n−3 +1 = 2 n 1 = − 2n 2 = −1 2 ∞ < γmt (Cn ).
∞ γmt (Cn
− {v1 , v2 , v3 }) =
∞ γmt (Pn−3 )
∞− ∞+ ∞− Hence γmt (Cn ) = 3. Therefore γmt (Cn ) + γmt (Cn ) = 5.
2
∞+ ∞− (Wn ) nor γmt (Wn ) exists. Remark 3.16 For Wheels Wn , n < 6, neither γmt ∞+ ∞− (Wn ) + γmt (Wn ) = 1, n ≥ 6. Theorem 3.17 For Wheels Wn , γmt
Proof. Let V (Wn ) = {v, v1 , v2 , . . . , vn } where deg(v) = n−1 and deg(vi ) = 3, ∞− ∞ 1 ≤ i ≤ n − 1. Clearly γmt (Wn ) = 2. Hence γmt (Wn ) does not exist. Now ∞+ ∞+ we find the value of γmt (Wn ). For n < 6, γmt (Wn ) does not exist. When ∞ n = 6, removal of v from Wn , results in a cycle Cn and γmt (C5 ) = 3. When ∞ n ≥ 7, removal of v from Wn , results in a cycle Cn−1 and γmt (Cn−1 ) ≥ 3. ∞+ ∞+ ∞− Hence γmt (Wn ) = 1. Hence γmt (Wn ) + γmt (Wn ) = 1. 2
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