On the broadcast domination number of permutation graphs

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On the broadcast domination number of permutation graphs Eunjeong Yi Texas A&M University at Galveston, Galveston, TX 77553, USA

a r t i c l e

i n f o

Article history: Received 23 April 2018 Received in revised form 3 March 2019 Accepted 19 March 2019 Available online xxxx Keywords: Distance Broadcast domination Permutation graph Generalized prism Cycles Paths Complete multipartite graphs

a b s t r a c t Broadcast domination models the idea of covering a network of cities by transmitters of varying powers while minimizing the total cost of the transmitters used to achieve full coverage. To be exact, let G be a connected graph of order at least two with vertex set V (G ) and edge set E (G ). Let d(x, y ), e ( v ), and diam(G ), respectively, denote the length of a shortest x − y path in G, the eccentricity of a vertex v in G, and the diameter of G. A function f : V (G ) → {0, 1, . . . , diam(G )} is called a broadcast if f ( v ) ≤ e ( v ) for each v ∈ V (G ). A broadcast f is called a dominating broadcast of G if, for each vertex u ∈ V (G ), there exists a vertex v ∈ V (G ) such that f ( v ) > 0 and d(u , v ) ≤ f ( v ). The broadcast domination  number, γb (G ), of G is the minimum of v ∈ V (G ) f ( v ) over all dominating broadcasts f on G. Let G 1 and G 2 be two disjoint copies of a graph G, and let σ : V (G 1 ) → V (G 2 ) be a bijection. Then a permutation graph G σ = ( V , E ) has vertex set V = V (G 1 ) ∪ V (G 2 ) and edge set E = E (G 1 ) ∪ E (G 2 ) ∪ {uv : v = σ (u )}. For a connected graph G of order at least two, we prove the sharp bounds 2 ≤ γb (G σ ) ≤ 2γb (G ); we give an example showing that there is no function h such that γb (G ) < h(γb (G σ )) for all pairs (G , σ ). We characterize G σ satisfying γb (G σ ) = 2, and examine γb (G σ ) when G is a cycle, a path, or a complete multipartite graph. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Let G = ( V (G ), E (G )) be a finite, simple, undirected, and connected graph of order at least two with vertex set V (G ) and edge set E (G ). The distance between two vertices x, y ∈ V (G ), denoted by d G (x, y ), is the length of a shortest path between x and y in G; the distance between a vertex v ∈ V (G ) and a set S ⊆ V (G ) is defined by d G ( v , S ) = min{d G ( v , x) : x ∈ S }. The eccentricity, e G ( v ), of a vertex v ∈ V (G ) is max{d G ( v , x) : x ∈ V (G )}; we drop the subscript G if it is clear from the context. The radius, rad(G ), of G is min{e ( v ) : v ∈ V (G )} and the diameter, diam(G ), of G is max{e ( v ) : v ∈ V (G )}. A vertex u ∈ V (G ) with e (u ) = rad(G ) is called a central vertex of G. The open neighborhood of a vertex v ∈ V (G ) is N G ( v ) = {u ∈ V (G ) : uv ∈ E (G )}, and its closed neighborhood is N G [ v ] = N G ( v ) ∪ { v }. For a set S ⊆ V (G ), its open neighborhood is the set N G ( S ) = ∪ v ∈ S N G ( v ) and its closed neighborhood is the set N G [ S ] = N G ( S ) ∪ S. More generally, for v ∈ V (G ), let N kG ( v ) = {u ∈

V (G ) : d G (u , v ) = k} and let N kG [ v ] = {u ∈ V (G ) : d G (u , v ) ≤ k}. The degree, degG ( v ), of a vertex v ∈ V (G ) is | N G ( v )|; a leaf is a vertex of degree one, and a support vertex is a vertex that is adjacent to a leaf. The maximum degree among the vertices of G is denoted by (G ). We denote by C n and P n , respectively, the cycle and the path on n vertices. A set S ⊆ V (G ) is a dominating set of G if every vertex in V (G ) − S is adjacent to at least one vertex of S, and the domination number, γ (G ), of G is the minimum cardinality over all dominating sets of G; a γ (G )-set is a minimum dominating set of G. As an application of domination in communication networks, Liu [13] provided a broadcast model, where a

E-mail address: [email protected]. https://doi.org/10.1016/j.tcs.2019.03.025 0304-3975/© 2019 Elsevier B.V. All rights reserved.

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dominating set represents cities with broadcast stations and its neighboring cities receive messages from the broadcast stations. Erwin [6,7] introduced broadcast domination that can be viewed as a generalization of Liu’s model, where cities with broadcast stations have transmission power that enable them to broadcast messages to cities at distances greater than one (depending on the transmission power of broadcast stations). A function f : V (G ) → {0, 1, . . . , diam(G )} is called a broadcast if f ( v ) ≤ e ( v ) for each v ∈ V (G ). A broadcast f is called a dominating broadcast of G if, for each vertex u ∈ V (G ), there exists a vertex  v ∈ V (G ) such that f ( v ) > 0 and d G (u , v ) ≤ f ( v ). The cost of a dominating broadcast fon a set S ⊆ V (G ) is c f ( S ) = v ∈ S f ( v ). The broadcast domination number, γb (G ), of G is the minimum of c f ( V (G )) = v ∈ V (G ) f ( v ), as f varies over all dominating broadcasts of G. It is known that determining the domination number of a general graph is an NP-complete problem (see [8]). Regarding the algorithmic aspect of broadcast domination, see [2,11]. For a given network, one naturally extends it to a larger network by considering two copies of the original network and a one-to-one correspondence between them; this may be represented as a permutation graph of the original graph. We refer to [1] for applications of the permutation graphs in a network problem. Chartrand and Harary [3] introduced a permutation graph (also called a generalized prism). Let G 1 and G 2 be two disjoint copies of a graph G, and let σ : V (G 1 ) → V (G 2 ) be a bijection. Then, a permutation graph G σ = ( V , E ) consists of vertex set V = V (G 1 ) ∪ V (G 2 ) and edge set E = E (G 1 ) ∪ E (G 2 ) ∪ {uv : v = σ (u )}. Notice that | V (G σ )| = 2| V (G )| and | E (G σ )| = 2| E (G )| + | V (G )|. Throughout this paper, we assume that G 1 and G 2 are two disjoint copies of G, σ is a bijection from V (G 1 ) to V (G 2 ), and σ −1 (the inverse of σ ) is a bijection from V (G 2 ) to V (G 1 ). This paper is organized as follows. In section 2, we recall some known results on (broadcast) domination and permutation graphs. In section 3, we prove the sharp bounds 2 ≤ γb (G σ ) ≤ 2γb (G ) for a connected graph G of order at least two; we give an example showing that there is no function h such that γb (G ) < h(γb (G σ )) for all pairs (G , σ ). In section 4, we characterize G σ satisfying γb (G σ ) = 2. In section 5, we characterize G σ such that γb (G σ ) equals 2 and 2γb (G ), respectively, when G is a cycle, a path, or a complete multipartite graph. 2. Preliminaries We begin by recalling some well-known results on the (broadcast) domination number of graphs. Observation 2.1. For n ≥ 3, γ ( P n ) = γ (C n ) = n3 . Theorem 2.2. [6,7] For a connected graph G of order at least two,



diam(G ) + 1 3



≤ γb (G ) ≤ min{rad(G ), γ (G )}.

Theorem 2.3. [6,7] Let G be a connected graph of order at least two. Then (a) (b)

γb (G ) = 1 if and only if rad(G ) = 1; γb (G ) = 2 if and only if min{rad(G ), γ (G )} = 2. As an immediate consequence of Observation 2.1 and Theorem 2.2, we have the following

Corollary 2.4. [6,7] For n ≥ 2, γb ( P n ) = n3 . Next, we recall the broadcast domination number of cycles. Proposition 2.5. [6] For n ≥ 3, γb (C n ) = n3 . Now, we recall the broadcast domination number of P m  P n , the Cartesian product of two paths. Note that P m  P 2 can be viewed as a permutation graph G σ on G = P m with σ : V (G 1 ) → V (G 2 ) mapping each vertex in G 1 to its corresponding vertex in G 2 ; we call such a map σ the identity map, and denote it by I . Theorem 2.6. [4] For m ≥ n ≥ 2, γb ( P m  P n ) = rad( P m  P n ) =  m +  n2 . 2 Next, we recall some results on permutation graphs. We begin with the following bounds of the domination number of permutation graphs. Proposition 2.7. [5,10] For every connected graph G and every bijection σ : V (G 1 ) → V (G 2 ),

γ (G ) ≤ γ (G σ ) ≤ 2γ (G ).

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Theorem 2.8. [9] Let G 1 and G 2 be two disjoint copies of a connected graph G, and let σ : V (G 1 ) → V (G 2 ) be a bijection. (a) If σ = I is the identity map, then rad(G I ) = rad(G ) + 1 and diam(G I ) = diam(G ) + 1. (b) For any  , 0 <  < 1, there exists a graph G satisfying diam(G σ ) ≤  · diam(G ). As an immediate consequence of Theorem 2.2, Proposition 2.7, and Theorem 2.8(a), we have the following Corollary 2.9. Let G 1 and G 2 be two disjoint copies of a connected graph G of order at least two, and let I : V (G 1 ) → V (G 2 ) be the identity map. Then



diam(G ) + 2 3

 ≤ γb (G I ) ≤ min{rad(G ) + 1, 2γ (G )}.

Remark 2.10. (a) If G = P 3 , then G I ∼ = P 3  P 2 and the lower bound of Corollary 2.9 holds. (b) If G = P 6 , then G I ∼ = P 6  P 2 and the upper bound of Corollary 2.9 holds. diam(G )+2

Proof. (a) By Theorem 2.6, γb (G I ) = 2 =

, since diam(G ) = 2. 3 (b) By Theorem 2.6, γb (G I ) = 4 = min{rad(G ) + 1, 2γ (G )}, since rad(G ) = 3 and

γ (G ) = 2. 2

3. General results on permutation graphs In this section, we prove that 2 ≤ γb (G σ ) ≤ 2γb (G ) for a connected graph G of order at least two; we give examples showing the sharpness of the bounds. We also provide an example showing that there is no function h such that γb (G ) < h(γb (G σ )) for all pairs (G , σ ). First, we obtain the bounds of the broadcast domination number of permutation graphs; here, we note that, unlike the case of domination (cf. Proposition 2.7), γb (G σ ) cannot be bounded below by a constant multiple of γb (G ) (see Remark 3.4). Proposition 3.1. Let G be a connected graph of order at least two. Let G 1 and G 2 be two disjoint copies of G, and let V (G 2 ) be a bijection. Then

σ : V (G 1 ) →

2 ≤ γb (G σ ) ≤ 2γb (G ), and the bounds are sharp. Proof. The lower bound follows from Theorem 2.3(a), since rad(G σ ) = 1 for any bijection σ . Next, we prove the upper bound. Let f : V (G ) → {0, 1, . . . , diam(G )} be a minimum dominating broadcast of G; then γb (G ) = c f ( V (G )). For each i ∈ {1, 2}, let f i : V (G i ) → {0, 1, . . . , diam(G i )} be a function corresponding to f on G i . Let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g ( v ) = min{ f i ( v ), e G σ ( v )} for v ∈ V (G i ), where i ∈ {1, 2}. Clearly, g ( v ) ≤ e G σ ( v ) for each v ∈ V (G σ ). For i ∈ {1, 2} and for each u ∈ V (G i ), noting that f i is a dominating broadcast of G i , there exists v ∈ V (G i ) such that g ( v ) > 0 satisfying (i) d G σ (u , v ) ≤ e G σ ( v ) = g ( v ) if f i ( v ) ≥ e G σ ( v ), or (ii) d G σ (u , v ) ≤ f i ( v ) = g ( v ) if f i ( v ) ≤ e G σ ( v ). Thus, g is a dominating broadcast of G σ and γb (G σ ) ≤ c g ( V (G 1 )) + c g ( V (G 2 )) ≤ c f 1 ( V (G 1 )) + c f 2 ( V (G 2 )) = 2c f ( V (G )) = 2γb (G ). If γb (G ) = 1, then, by the current proposition, γb (G σ ) = 2 for an arbitrary bijection σ ; thus, G σ achieves both the upper and lower bounds. For the sharpness of the lower bound when γb (G ) = 1, see Remark 3.2; for the sharpness of the upper bound when γb (G ) = 1, see Remark 3.3. 2 Remark 3.2. There exists a permutation graph G σ satisfying

γb (G σ ) = 2 = 2γb (G ).

Proof. Let G be the graph obtained from a star K 1,x on x + 1 vertices, where x ≥ 3, by subdividing every edge exactly once; then rad(G ) = 2, and hence γb (G ) = 2 by Theorem 2.3. Let V (G 1 ) = {u 0 } ∪ (∪ix=1 {u i ,1 , u i ,2 }), where u 0 is the central vertex of degree x, u i ,1 is a vertex of degree two that is adjacent to u 0 and a leaf u i ,2 in G 1 ; similarly, let V (G 2 ) = { w 0 } ∪ (∪ix=1 { w i ,1 , w i ,2 }), where w 0 is the central vertex of degree x, w i ,1 is a vertex of degree two that is adjacent to w 0 and a leaf w i ,2 in G 2 . Let σ : V (G 1 ) → V (G 2 ) be a bijection given by σ (u 0 ) = w 0 , σ (u i ,1 ) = w i ,2 and σ (u i ,2 ) = w i ,1 for each i ∈ {1, 2, . . . , x}, where x ≥ 3. Then rad(G σ ) = 2, and thus γb (G σ ) = 2 by Theorem 2.3. 2 Remark 3.3. There exists a permutation graph G σ satisfying

γb (G σ ) = 2γb (G ) = 2.

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Fig. 1. Permutation graph G σ satisfying

γb (G σ ) = 2γb (G ) = 2.

Proof. Let G be the graph obtained from a disjoint union of two stars K 1,x and K 1, y , where x, y ≥ 3, by adding an y edge between a leaf of K 1,x and a leaf of K 1, y . Let V (G 1 ) = {s1 , s2 } ∪ (∪ix=1 {u 1,i }) ∪ (∪ j =1 {u 2, j }), where degG 1 (s1 ) = x, degG 1 (s2 ) = y, degG 1 (u 1,x ) = 2 = degG 1 (u 2, y ), and degG 1 (u 1,i ) = degG 1 (u 2, j ) = 1 for each i ∈ {1, 2, . . . , x − 1} and for y

each j ∈ {1, 2, . . . , y − 1}; similarly, let V (G 2 ) = {t 1 , t 2 } ∪ (∪ix=1 { w 1,i }) ∪ (∪ j =1 { w 2, j }), where degG 2 (t 1 ) = x, degG 2 (t 2 ) = y, degG 2 ( w 1,x ) = 2 = degG 2 ( w 2, y ), and degG 2 ( w 1,i ) = degG 2 ( w 2, j ) = 1 for each i ∈ {1, 2, . . . , x − 1} and for each j ∈ {1, 2, . . . , y − 1}. Let σ : V (G 1 ) → V (G 2 ) be a bijection given by σ (si ) = t i for i ∈ {1, 2}, σ (u 1,i ) = w 1,i and σ (u 2, j ) = w 2, j for each i ∈ {1, 2, . . . , x} and for each j ∈ {1, 2, . . . , y }, where x, y ≥ 3. See Fig. 1. First, we show that γb (G ) = 2. Since rad(G ) = 1 and S = {s1 , s2 } forms a dominating set of G 1 with | S | = 2, γb (G ) = γb (G 1 ) = 2 by Theorem 2.3. Next, we show that γb (G σ ) = 4 = 2γb (G ). Since S = {s1 , s2 , t 1 , t 2 } forms a dominating set of G σ with | S | = 4, γb (G σ ) ≤ 4 by Theorem 2.2. We show that γb (G σ ) ≥ 4. Since diam(G σ ) = 6, γb (G σ ) ≥ 3 by Theorem 2.2. Let A 1 = y y {s1 } ∪ (∪ix=1 {u 1,i }), A 2 = {s2 } ∪ (∪ j =1 {u 2, j }), A 3 = {t 1 } ∪ (∪ix=1 { w 1,i }), and A 4 = {t 2 } ∪ (∪ j =1 { w 2, j }). Assume that f : V (G σ ) → {1, 2, . . . , diam(G σ )} is a dominating broadcast of G σ with c f ( V (G σ )) = 3; then c f ( A i ) = 0 for some i ∈ {1, 2, 3, 4}. Without loss of generality, assume that c f ( A 1 ) = 0. If c f ( A 1 ∪ A 2 ∪ A 3 ) = 0, then c f ( A 4 ) ≥ 4 since d G σ (u 1,1 , A 4 ) = 4; thus, c f ( A i ) = 0 for at most two i ∈ {1, 2, 3, 4}. If c f ( A 1 ∪ A 2 ) = 0, then c f ( A 3 ∪ A 4 ) ≥ 4: if c f ( A 3 ) = 2 and c f ( A 4 ) = 1, or c f ( A 3 ) = 1 and c f ( A 4 ) = 2, say the former, then c f (t 1 ) = 2 in order for every vertex in A 1 ∪ A 3 to be at distance at most two from a vertex in A 3 , and at most one vertex in A 2 can be at distance one from a vertex in A 4 , and hence f fails to be a dominating broadcast of G σ . If c f ( A 1 ∪ A 3 ) = 0, then c f ( A 2 ∪ A 4 ) ≥ 4 since d G σ (u 1,1 , A 2 ) = 3 = d G σ ( w 1,1 , A 4 ) and d G σ (u 1,1 , A 4 ) = 4 = d G σ ( w 1,1 , A 2 ). If c f ( A 1 ∪ A 4 ) = 0, then c f ( A 2 ∪ A 3 ) ≥ 4: if c f ( A 2 ) = 2 and c f ( A 3 ) = 1, or c f ( A 2 ) = 1 and c f ( A 3 ) = 2, say the former, then c f (s2 ) = 2 in order for every vertex in A 2 ∪ A 4 to be at distance at most two from a vertex in A 2 , and exactly one vertex in A 1 is within distance two from s2 ∈ A 2 and at most one vertex in A 1 can be at distance one from a vertex in A 3 , and hence f fails to be a dominating broadcast of G σ . Now, suppose that c f ( A i ) = 0 for exactly one i ∈ {1, 2, 3, 4}; then c f ( A 2 ) = c f ( A 3 ) = c f ( A 4 ) = 1, and at most one vertex in A 1 can be at distance one from a vertex in A 2 ( A 3 , respectively), and hence f fails to be a dominating broadcast of G σ . So, c f ( V (G σ )) ≥ 4 in each case, and thus γb (G σ ) = 4. 2 Based on Theorem 2.8(b), it is natural to consider an example such that γb (G ) − γb (G σ ) can be arbitrarily large. We will provide an example showing that there is no function h such that γb (G ) < h(γb (G σ )) for all pairs (G , σ ). Remark 3.4. There exists a permutation graph G σ such that function h such that γb (G ) < h(γb (G σ )) for all pairs (G , σ ).

γb (G ) − γb (G σ ) is arbitrarily large; moreover, there is no

Proof. Let G be a graph obtained by attaching 3k leaves to a support vertex of P 3k+1 , where k ≥ 3. Let G 1 be the graph obtained from P 3k+1 , given by u 1 , u 2 , . . . , u 3k+1 , by attaching 3k leaves, say x1 , x2 , . . . , x3k , to the vertex u 2 ; similarly, let G 2 be the graph obtained from P 3k+1 , given by w 1 , w 2 , . . . , w 3k+1 , by attaching 3k leaves, say y 1 , y 2 , . . . , y 3k , to the vertex w 2 (see Fig. 2). Let σ : V (G 1 ) → V (G 2 ) be a bijection given by σ (u 1 ) = y 1 , σ (u 2 ) = w 2 , σ (u i ) = y i −1 for i ∈ {3, 4, . . . , 3k + 1}, σ (x1 ) = w 1 , and σ (x j ) = w j+1 for j ∈ {2, 3, . . . , 3k}. First, we show that γb (G ) = k + 1. Since diam(G ) = diam( P 3k+1 ) = 3k, γb (G ) ≥ k + 1 by Theorem 2.2. On the other −1 hand, S = (∪ki = 0 {u 3i +2 }) ∪ {u 3k+1 } forms a dominating set of G 1 with | S | = k + 1, and hence γb ( G ) = γb ( G 1 ) ≤ k + 1 by Theorem 2.2. Thus, γb (G ) = k + 1.

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Fig. 2. Permutation graph G σ showing that there is no function h such that

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γb (G ) < h(γb (G σ )) for all pairs (G , σ ).

Second, we show that γb (G σ ) = 3. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f (u 2 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 2 }, then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3; thus, γb (G σ ) ≤ 3. On the other hand, γb (G σ ) ≥ 3 by Theorem 2.3, since γ (G σ ) ≥ 3 and rad(G σ ) ≥ 3. Thus, γb (G σ ) = 3. Therefore, γb (G ) − γb (G σ ) = k − 2 can be arbitrarily large; furthermore, there is no function h such that γb (G ) < h(γb (G σ )) for all pairs (G , σ ). 2 Remark 3.5. The concept of distance k-domination, for k ≥ 1, was introduced by Meir and Moon [14]. A set of vertices S ⊆ V (G ) is a distance k-dominating set of G if, for each u ∈ V (G ) − S, d G (u , S ) ≤ k, and the distance k-domination number of G, denoted by γk (G ), is the minimum cardinality over all distance k-dominating sets of G. Hurtado et al. [12] studied distance 2-domination number of permutation graphs. Remark 3.4 is implied by Theorem 4 of [12], since γb (G σ ) ≤ 2γ2 (G σ ) = 4. 4. Permutation graphs G σ satisfying γb ( G σ ) = 2 In this section, we characterize permutation graphs G σ for which its broadcast domination number equals two. By Theorem 2.3(b), it suffices to characterize permutation graphs G σ satisfying γ (G σ ) = 2 and rad(G σ ) = 2, respectively. First, we examine permutation graphs G σ with γ (G σ ) = 2. Proposition 4.1. Let G be a connected graph of order at least three, and let G 1 and G 2 be two disjoint copies of G. Let σ : V (G 1 ) → V (G 2 ) be a bijection. Then γ (G σ ) = 2 if and only if (a) (G ) = | V (G )| − 1 and σ is an arbitrary bijection, or (b) there exist two vertices u ∈ V (G 1 ) and w ∈ V (G 2 ) satisfying d G 1 (u , u  ) = 2 = d G 2 ( w , w  ), N G 1 [u ] ∪ {u  } = V (G 1 ), and N G 2 [ w ] ∪ { w  } = V (G 2 ) such that σ (u  ) = w and σ (u ) = w  . Proof. Let σ : V (G 1 ) → V (G 2 ) be a bijection. Notice that γ (G σ ) ≥ 2, since (G σ ) = | V (G σ )| − 1 for | V (G )| ≥ 3. (⇐) (a) Let (G ) = | V (G )| − 1. Then there exist u ∈ V (G 1 ) and w ∈ V (G 2 ) such that degG 1 (u ) = | V (G )| − 1 = degG 2 ( w ). Since N G σ [{u , w }] = V (G 1 ) ∪ V (G 2 ) = V (G σ ), {u , w } is a γ (G σ )-set for an arbitrary bijection σ ; thus, γ (G σ ) = 2. (b) Let u ∈ V (G 1 ) and w ∈ V (G 2 ) satisfy the following conditions: d G 1 (u , u  ) = 2 = d G 2 ( w , w  ), N G 1 [u ] ∪ {u  } = V (G 1 ), N G 2 [ w ] ∪{ w  } = V (G 2 ), σ (u  ) = w, and σ (u ) = w  . Then N G σ [u ] = ( V (G 1 ) −{u  }) ∪{ w  } and N G σ [ w ] = ( V (G 2 ) −{ w  }) ∪{u  }; thus, N G σ [{u , w }] = V (G 1 ) ∪ V (G 2 ), and hence γ (G σ ) = 2. (⇒) Let S be a γ (G σ )-set with | S | = 2; let S 1 = S ∩ V (G 1 ) and S 2 = S ∩ V (G 2 ). If | S 1 | = 2 or | S 2 | = 2, say the former, then exactly two vertices in V (G 2 ) are dominated by S 1 ; thus, S 1 fails to be a dominating set of G σ , since | V (G 2 )| = | V (G )| ≥ 3. So, we must have | S 1 | = | S 2 | = 1. Notice that | N G 1 [ S 1 ]| ≥ | V (G 1 )| − 1; otherwise, at least two vertices in V (G 1 ) − N G 1 [ S 1 ] must be dominated by S 2 , which is impossible since | V (G 1 ) − N G 1 [ S 1 ]| ≥ 2 and | S 2 | = 1. Similarly, | N G 2 [ S 2 ]| ≥ | V (G 2 )| − 1. First, let | N G 1 [ S 1 ]| = | V (G 1 )|. Then (G ) = | V (G )| − 1, and thus satisfying (a) of the current proposition. Second, let | N G 1 [ S 1 ]| = | V (G 1 )| − 1. Further, let (G ) = | V (G )| − 1; then | N G 2 [ S 2 ]| = | V (G 2 )| − 1. Suppose that S 1 = {u }, S 2 = { w }, V (G 1 ) − N G 1 [ S 1 ] = {u  }, and V (G 2 ) − N G 2 [ S 2 ] = { w  }. Then d G 1 (u , u  ) = 2, d G 2 ( w , w  ) = 2, N G 1 [u ] ∪ {u  } = V (G 1 ), and N G 2 [ w ] ∪ { w  } = V (G 2 ). Moreover, in order for u  to be dominated by w ∈ S 2 , σ (u  ) = w; similarly, in order for w  to be dominated by u ∈ S 1 , σ −1 ( w  ) = u. So, G σ satisfies the condition (b) of the current proposition. 2

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Next, we examine permutation graphs G σ satisfying rad(G σ ) = 2. Lemma 4.2. Let G 1 and G 2 be two disjoint copies of a connected graph G of order at least three, and let bijection. If rad(G σ ) = 2, then rad(G ) ∈ {1, 2}.

σ : V (G 1 ) → V (G 2 ) be a

Proof. Suppose that rad(G ) = k ≥ 3. Then, for each vertex u ∈ V (G 1 ), e G 1 (u ) ≥ rad(G 1 ) = k ≥ 3. Note that, for each vertex x ∈ V (G 1 ) − N G2 1 [u ], d G 1 (u , x) ≥ 3 and d G σ (u , x) ≥ 3; thus e G σ (u ) ≥ 3 for each u ∈ V (G 1 ), and hence rad(G σ ) ≥ 3. Therefore, rad(G σ ) = 2 implies rad(G ) ∈ {1, 2}. 2 Proposition 4.3. Let G be a connected graph of order at least three, and let G 1 and G 2 be two disjoint copies of G. Let σ : V (G 1 ) → V (G 2 ) be a bijection. Then rad(G σ ) = 2 if and only if (a) rad(G ) = 1 (equivalently, (G ) = | V (G )| − 1), or (b) rad(G ) = 2 and there exists a vertex u ∈ V (G 1 ) or u ∈ V (G 2 ), say the former, with e G 1 (u ) = rad(G 1 ) = 2 such that σ −1 ( V (G 2 ) − N G 2 [σ (u )]) ⊆ N G 1 (u ). Proof. (⇐) (a) Let rad(G ) = 1, i.e., (G ) = | V (G )| − 1. Then there exists u ∈ V (G 1 ) with e G 1 (u ) = rad(G 1 ) = 1 and e G σ (u ) = 2, since N G2 σ [u ] = V (G σ ) and N G σ [u ] = V (G σ ); thus rad(G σ ) ≤ 2. Noting that (G σ ) = | V (G σ )| − 1 (i.e., rad(G σ ) = 1) for | V (G )| ≥ 3, we have rad(G σ ) = 2. (b) Let rad(G ) = 2. Suppose that u ∈ V (G 1 ) such that e G 1 (u ) = rad(G 1 ) = 2 and σ −1 ( V (G 2 ) − N G 2 [σ (u )]) ⊆ N G 1 (u ). Then 2 N G σ [u ] = V (G σ ) and N G σ [u ] = V (G σ ); thus, rad(G σ ) = 2. (⇒) Let rad(G σ ) = 2. By Lemma 4.2, rad(G ) ∈ {1, 2}. If rad(G ) = 1 ⇔ (G ) = | V (G )| − 1, then (a) of the current proposition is satisfied. So, suppose that rad(G ) = 2. Then there exists a vertex u ∈ V (G 1 ) or u ∈ V (G 2 ), say the former, with e G 1 (u ) = rad(G 1 ) = 2, i.e., N G2 1 [u ] = V (G 1 ) and N G 1 [u ] = V (G 1 ). In order for G σ to satisfy rad(G σ ) = 2, we must have

σ −1 ( V (G 2 ) − N G 2 [σ (u )]) ⊆ N G 1 (u ). So, G σ satisfies (b) of the current proposition. 2

We note that if G is a connected graph of order two, then G ∼ = P 2 and G σ ∼ = C 4 satisfying γb (G σ ) = 2 = 2γb (G ) for any bijection σ ; here, G satisfies (a) of Propositions 4.1 and 4.3. So, Theorem 2.3, combined together with Propositions 4.1 and 4.3, implies the following Theorem 4.4. Let G 1 and G 2 be two disjoint copies of a connected graph G of order at least two. Let σ : V (G 1 ) → V (G 2 ) be a bijection. Then γb (G σ ) = 2 if and only if one of the followings holds: (a) (G ) = | V (G )| − 1 and σ is an arbitrary bijection; (b) there exist two vertices u ∈ V (G 1 ) and w ∈ V (G 2 ) satisfying d G 1 (u , u  ) = 2 = d G 2 ( w , w  ), N G 1 [u ] ∪ {u  } = V (G 1 ), and N G 2 [ w ] ∪ { w  } = V (G 2 ) such that σ (u  ) = w and σ (u ) = w  ; (c) rad(G ) = 2 and there exists a vertex v ∈ V (G 1 ) or v ∈ V (G 2 ), say the former, with e G 1 ( v ) = rad(G 1 ) = 2 such that σ −1 ( V (G 2 ) − N G 2 [σ ( v )]) ⊆ N G 1 ( v ). 5. Permutation graphs on cycles, paths, and complete multipartite graphs In this section, we examine some permutation graphs achieving the lower bound or the upper bound of Proposition 3.1. We begin with permutation graphs on cycles. First, we recall the following result. Theorem 5.1. [5] Let G 1 and G 2 be two disjoint copies of G = C n for n ≥ 3, and let σ : V (G 1 ) → V (G 2 ) be a bijection. (a) If n ≡ 1 (mod 3) or n ≡ 2 (mod 3), then γ (G σ ) < 2γ (G ). (b) If n ≡ 0 (mod 3), then there exists G σ satisfying γ (G σ ) = 2γ (G ) for each n ≥ 9. In contrast to the case of domination, we prove the following result on broadcast domination. Theorem 5.2. Let G 1 and G 2 be two disjoint copies of G = C n for n ≥ 3, and let σ : V (G 1 ) → V (G 2 ) be a bijection. Then 2 ≤ γb (G σ ) ≤ 2γb (G ) = 2 n3 . Moreover, we have the following: (I) (II)

γb (G σ ) = 2 if and only if G σ is isomorphic to one of the permutation graphs in Fig. 3; γb (G σ ) = 2γb (G ) if and only if G = C 3 and σ is an arbitrary bijection.

Proof. The bounds of γb (G σ ) follow from Propositions 2.5 and 3.1. Let V (G 1 ) = {u 1 , u 2 , . . . , un }, V (G 2 ) = { w 1 , w 2 , . . . , w n }, E (G 1 ) = {u i u i +1 : 1 ≤ i ≤ n − 1} ∪ {un u 1 }, and E (G 2 ) = { w i w i +1 : 1 ≤ i ≤ n − 1} ∪ { w n w 1 }, where n ≥ 3.

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Fig. 3. Permutation graphs G σ satisfying

7

γb (G σ ) = 2 for G = C n , n ≥ 3.

(I) (⇐) If G σ is isomorphic to Fig. 3(a) or 3(b), then γ (G σ ) = 2 since the solid vertices form a γ (G σ )-set; thus γb (G σ ) = 2 by Theorem 2.3. If G σ is isomorphic to Fig. 3(c), 3(d), or 3(e), then rad(G σ ) = 2 since e G σ (x) = 2; thus γb (G σ ) = 2 by Theorem 2.3. (⇒) Let γb (G σ ) = 2. If n ≥ 6, then rad(G ) ≥ 3; thus G σ fails to satisfy any one of the three conditions of Theorem 4.4. So, n ∈ {3, 4, 5}. If n = 3, then G σ is isomorphic to Fig. 3(a) and (G ) = | V (G )| − 1, satisfying (a) of Theorem 4.4. If n = 4, then G σ is isomorphic to (b) or (c) of Fig. 3 (see [3]): if G σ is isomorphic to Fig. 3(b), then G σ satisfies (b) of Theorem 4.4; if G σ is isomorphic to Fig. 3(c), then G σ satisfies (c) of Theorem 4.4. Next, let n = 5; then rad(G ) = 2. Clearly, G σ fails to satisfy (a) or (b) of Theorem 4.4. Noting that e G 1 (u 1 ) = 2, we may assume that σ (u 1 ) = w 1 by relabeling if necessary. In order for G σ to satisfy (c) of Theorem 4.4, {σ (u 2 ), σ (u 5 )} = { w 3 , w 4 }; we may assume that σ (u 2 ) = w 3 and σ (u 5 ) = w 4 by relabeling if necessary. So, {σ (u 3 ), σ (u 4 )} = { w 2 , w 5 }. If σ (u 3 ) = w 2 , then G σ is isomorphic to Fig. 3(d); if σ (u 3 ) = w 5 , then G σ is isomorphic to Fig. 3(e). (II) If n ≡ 1, 2 (mod 3), then γb (G σ ) ≤ γ (G σ ) < 2γ (G ) = 2γb (G ) by Theorem 2.2, Theorem 5.1(a), Observation 2.1 and Proposition 2.5. So, let n ≡ 0 (mod 3); we show that γb (G σ ) = 2γb (G ) if and only if G = C 3 and σ is an arbitrary bijection. If n = 3, then γb (G σ ) = 2 = 2γb (G ) by (a) of Theorems 2.3 and 4.4. Now, let n = 3k = 3; then γb (G ) = k by Proposition 2.5. Let σ (u 1 ) = w 1 by relabeling if necessary. Case 1: σ (u 3a+1 ) = w 3b+1 for some a, b ∈ {1, 2, . . . , k − 1}. Let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function given by

⎧ ⎨ 2 if v = u 3a+1 , −1 1 {u 3i }) ∪ (∪ki =−a1+1 {u 3i +2 }) ∪ [(∪kj− f ( v ) = 1 if v ∈ (∪ai = 1 =0 { w 3 j +1 }) − { w 3b+1 }], ⎩ 0 otherwise. Then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2 + (a − 1) + (k − 1 − a) + (k − 1) = 2k − 1; thus γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Case 2: σ (u 3a+1 ) = w 3b+2 for some a ∈ {1, 2, . . . , k − 1} and for some b ∈ {0, 1, . . . , k − 1}. Let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function: (i) if b = k − 1, then let g be defined by

⎧ ⎨ 3 if v = σ (u 3a+1 ) = w 3b+2 , −1 1 {u 3i }) ∪ (∪ki =−a1+1 {u 3i +2 }) ∪ [(∪kj− g ( v ) = 1 if v ∈ (∪ai = 1 =0 { w 3 j +1 }) − { w 3b+1 , w 3b+4 }], ⎩ 0 otherwise; (ii) if b = k − 1, then let g be defined by

⎧ ⎨ 3 if v = σ (u 3a+1 ) = w 3b+2 , −1 2 {u 3i }) ∪ (∪ki =−a1+1 {u 3i +2 }) ∪ (∪kj − g ( v ) = 1 if v ∈ (∪ai = 1 =1 { w 3 j +1 }), ⎩ 0 otherwise. In each case, g is a dominating broadcast of G σ with c g ( V (G σ )) = 3 + (a − 1) + (k − 1 − a) + (k − 2) = 2k − 1. Thus, γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Case 3: σ (u 3a+1 ) = w 3b for some a ∈ {1, 2, . . . , k − 1} and for some b ∈ {1, 2, . . . , k}. One can easily check that each permutation graph G σ in Case 3 is isomorphic to some permutation graph belonging to Case 2, by fixing the labeling of G 1 and by relabeling the vertices of G 2 in the reverse cyclic order with w 1 being fixed. So, γb (G σ ) < 2γb (G ) in this case. Therefore, γb (G σ ) < 2γb (G ) for G = C 3k , where k ≥ 2. 2 Next, we consider permutation graphs on paths. We begin with the following lemma, which will be used in proving Theorem 5.4. Lemma 5.3. Let G 1 and G 2 be two disjoint copies of G = P 6 , and let and only if G σ ∼ = P 6 P 2.

σ : V (G 1 ) → V (G 2 ) be a bijection. Then γb (G σ ) = 2γb (G ) if

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Fig. 4. Permutation graphs ( P 6 )σ  P 6  P 2 satisfying σ (u 2 ) = w 2 and σ (u 5 ) = w 5 , where a dotted segment indicates non-adjacency and a positive integer value inscribed in a vertex denotes the value of a dominating broadcast of ( P 6 )σ at the vertex.

Fig. 5. Permutation graphs ( P 6 )σ satisfying

σ (u 2 ) = w 2 and σ (u 5 ) = w 5 .

Proof. Note that γb (G ) = 2 by Corollary 2.4. Let V (G 1 ) = {u 1 , u 2 , u 3 , u 4 , u 5 , u 6 }, V (G 2 ) = { w 1 , w 2 , w 3 , w 4 , w 5 , w 6 }, E (G 1 ) = {u i u i +1 : 1 ≤ i ≤ 5}, and E (G 2 ) = { w i w i +1 : 1 ≤ i ≤ 5}. Case 1: |{σ (u 2 ), σ (u 5 )} ∩ { w 2 , w 5 }| = 2. We may assume that σ (u 2 ) = w 2 and σ (u 5 ) = w 5 , by relabeling if necessary (see Fig. 4). First, let σ (u 6 ) ∈ / { w 4 , w 6 } (see Fig. 4(a)). If f : V (G σ ) → {0, 1, . . . , diam(G σ )} is a function defined by f (u 3 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 3 }, then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3; thus, γb (G σ ) ≤ 3 < 2γb (G ). Second, let σ (u 6 ) = w 4 (see Fig. 4(b)). If σ (u 4 ) = w 6 , let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f ( w 3 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − { w 3 }; then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3, and thus γb (G σ ) ≤ 3 < 2γb (G ). Now, let σ (u 4 ) = w 6 . If σ (u 1 ) = w 1 , let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g (u 3 ) = 3 and g ( v ) = 0 for each v ∈ V (G σ ) − {u 3 }; if σ (u 1 ) = w 3 , let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g (u 3 ) = 2, g ( w 4 ) = 1, and g ( v ) = 0 for each v ∈ V (G σ ) − {u 3 , w 4 }. In each case, g is a dominating broadcast of G σ with c g ( V (G σ )) = 3, and hence γb (G σ ) ≤ 3 < 2γb (G ). Third, let σ (u 6 ) = w 6 (see Fig. 4(c)). If σ (u 4 ) = w 4 , let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f (u 4 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 4 }; then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3, and thus γb (G σ ) ≤ 3 < 2γb (G ). So, let σ (u 4 ) = w 4 . If σ (u 1 ) = w 1 , then γb (G σ ) = γb ( P 6  P 2 ) = 4 = 2γb (G ) by Corollary 2.4 and Theorem 2.6. If σ (u 1 ) = w 3 , let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g (u 4 ) = 3 and g ( v ) = 0 for each v ∈ V (G σ ) − {u 4 }; then g is a dominating broadcast of G σ with c g ( V (G σ )) = 3, and hence γb (G σ ) ≤ 3 < 2γb (G ). Case 2: |{σ (u 2 ), σ (u 5 )} ∩ { w 2 , w 5 }| = 1. Let σ (u 2 ) = w 2 by relabeling if necessary; then σ (u 5 ) ∈ { w 1 , w 3 , w 4 , w 6 } (see Fig. 5). First, let σ (u 5 ) = w 1 (see Fig. 5(a)). If σ (u 3 ) = w 3 , let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f (u 3 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 3 }; if σ (u 3 ) = w 3 , let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f (u 4 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 4 }. In each case, f is a dominating broadcast of G σ with c f ( V (G σ )) = 3, and thus γb (G σ ) ≤ 3 < 2γb (G ). Second, let σ (u 5 ) ∈ { w 3 , w 4 } (see Fig. 5(b)). If f : V (G σ ) → {0, 1, . . . , diam(G σ )} is a function defined by f ( w 3 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − { w 3 }, then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3; thus, γb (G σ ) ≤ 3 < 2γb (G ). Third, let σ (u 5 ) = w 6 (see Fig. 5(c)). If σ (u 3 ) = w 1 , let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g (u 3 ) = 3 and g ( v ) = 0 for each v ∈ V (G σ ) − {u 3 }; if σ (u 3 ) = w 1 , let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g (u 4 ) = 3 and g ( v ) = 0 for each v ∈ V (G σ ) − {u 4 }. In each case, g is a dominating broadcast of G σ with c g ( V (G σ )) = 3, and hence γb (G σ ) ≤ 3 < 2γb (G ). Case 3: |{σ (u 2 ), σ (u 5 )} ∩ { w 2 , w 5 }| = 0. We consider three subcases.

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Fig. 6. Permutation graphs ( P 6 )σ satisfying {σ (u 2 ), σ (u 5 )} ∩ { w 2 , w 5 } = ∅.

Fig. 7. Permutation graphs G σ satisfying

γb (G σ ) = 2 for G = P n , n ≥ 2.

Subcase 3.1: |{σ (u 2 ), σ (u 5 )} ∩ { w 1 , w 6 }| = 2. Let σ (u 2 ) = w 1 and σ (u 5 ) = w 6 , by relabeling if necessary (see Fig. 6(a)). If σ (u 3 ) = w 2 , let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f (u 3 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 3 }; if σ (u 3 ) = w 2 , let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f (u 4 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − {u 4 }. In each case, f is a dominating broadcast of G σ with c f ( V (G σ )) = 3, and thus γb (G σ ) ≤ 3 < 2γb (G ). Subcase 3.2: |{σ (u 2 ), σ (u 5 )} ∩ { w 1 , w 6 }| = 1. By relabeling if necessary, let σ (u 2 ) = w 1 ; then σ (u 5 ) ∈ { w 3 , w 4 } (see Fig. 6(b)). First, let σ (u 5 ) = w 3 . If σ (u 4 ) = w 2 , then let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by g (u 4 ) = 3 and g ( v ) = 0 for each v ∈ V (G σ ) − {u 4 }; if σ (u 4 ) = w 2 , then let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

g (u 3 ) = 3 and g ( v ) = 0 for each v ∈ V (G σ ) − {u 3 }. In each case, g is a dominating broadcast of G σ with c g ( V (G σ )) = 3, and thus γb (G σ ) ≤ 3 < 2γb (G ). Second, let σ (u 5 ) = w 4 . If we let h : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by h(u 2 ) = 1, h( w 4 ) = 2, and h( v ) = 0 for each v ∈ V (G σ ) −{u 2 , w 4 }, then h is a dominating broadcast of G σ with ch ( V (G σ )) = 3; thus, γb (G σ ) ≤ 3 < 2γb (G ). Subcase 3.3: |{σ (u 2 ), σ (u 5 )} ∩ { w 1 , w 6 }| = 0. In this case, {σ (u 2 ), σ (u 5 )} = { w 3 , w 4 }; let σ (u 2 ) = w 3 and σ (u 5 ) = w 4 by relabeling if necessary (see Fig. 6(c)). If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by f ( w 3 ) = 3 and f ( v ) = 0 for each v ∈ V (G σ ) − { w 3 }, then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3; thus, γb (G σ ) ≤ 3 < 2γb (G ). 2 Theorem 5.4. Let G 1 and G 2 be two disjoint copies of G = P n for n ≥ 2, and let σ : V (G 1 ) → V (G 2 ) be a bijection. Then 2 ≤ γb (G σ ) ≤ 2γb (G ) = 2 n3 . Moreover, we have the following: (I) γb (G σ ) = 2 if and only if G σ is isomorphic to one of the permutation graphs in Fig. 7; (II) if n ≡ 1 (mod 3), then γb (G σ ) < 2γb (G ); (III) if n ≡ 2 (mod 3), then γb (G σ ) = 2γb (G ) if and only if G = P 2 and σ is an arbitrary bijection; (IV) if n ≡ 0 (mod 3), then γb (G σ ) = 2γb (G ) if and only if G = P 3 and σ is an arbitrary bijection, or G σ ∼ = P 6 P 2. Proof. The bounds of γb (G σ ) follow from Corollary 2.4 and Proposition 3.1. Let V (G 1 ) = {u 1 , u 2 , . . . , un }, V (G 2 ) = { w 1 , w 2 , . . . , w n }, E (G 1 ) = {u i u i +1 : 1 ≤ i ≤ n − 1}, and E (G 2 ) = { w i w i +1 : 1 ≤ i ≤ n − 1}, where n ≥ 2. (I) (⇐) If G σ is isomorphic to Fig. 7(a), 7(b), 7(c), 7(g), or 7(h), then γ (G σ ) = 2 since the solid vertices form a γ (G σ )-set; thus γb (G σ ) = 2 by Theorem 2.3. If G σ is isomorphic to Fig. 7(d), 7(e), 7(f), 7(i), 7(j), 7(k), 7(l), or 7(m), then rad(G σ ) = 2 since e G σ (x) = 2; thus γb (G σ ) = 2 by Theorem 2.3.

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(⇒) Suppose that γb (G σ ) = 2. If n ≥ 6, then G σ fails to satisfy any one of the three conditions of Theorem 4.4. So, n ∈ {2, 3, 4, 5}. First, let n = 2; then G σ is isomorphic to Fig. 7(a). Second, let n = 3; then G σ is isomorphic to Fig. 7(b) or 7(c). Third, let n = 4; then Theorem 4.4(a) fails. If G σ satisfies the condition (b) of Theorem 4.4, then G σ is isomorphic to Fig. 7(g) or 7(h). If G σ satisfies the condition (c) of Theorem 4.4, then G σ is isomorphic to Fig. 7(d), 7(e), 7(f), or 7(i). Next, suppose that n = 5; then, clearly, G σ fails to satisfy (a) or (b) of Theorem 4.4. Noting that e G 1 (u 3 ) = 2 < e G 1 (u i ) for i ∈ {1, 2, 4, 5} and that | V (G 2 ) − N G 2 [σ (u 3 )]| ≤ | N G 1 (u 3 )| = 2, it is necessary that σ (u 3 ) ∈ { w 2 , w 3 , w 4 } in order for G σ to satisfy the condition (c) of Theorem 4.4. If σ (u 3 ) = w 2 or σ (u 3 ) = w 4 , say the former, by relabeling if necessary, then G σ is isomorphic to Fig. 7(j) or 7(k). If σ (u 3 ) = w 3 , then G σ is isomorphic to Fig. 7(l) or 7(m). (II) Let n ≡ 1 (mod 3). We write n = 3k + 1, where k ≥ 1; then γb (G ) = k + 1 by Corollary 2.4. If σ (u 1 ) = w 3b+1 for some b ∈ {0, 1, . . . , k}, then S = (∪ki=1 {u 3i }) ∪ (∪kj=0 { w 3 j +1 }) forms a dominating set of G σ with | S | = 2k + 1; if σ (u 1 ) = w 3b+2

for some b ∈ {0, 1, . . . , k − 1}, then S = (∪ki=1 {u 3i }) ∪ (∪bj=0 { w 3 j +2 }) ∪ (∪kj=b+1 { w 3 j +1 }) forms a dominating set of G σ with −1 | S | = 2k + 1; if σ (u 1 ) = w 3b for some b ∈ {1, . . . , k}, then S = (∪ki=1 {u 3i }) ∪ (∪bj= { w 3 j+2 }) ∪ (∪kj=b { w 3 j }) forms a dominating 0 set of G σ with | S | = 2k + 1. In each case, γb (G σ ) ≤ γ (G σ ) ≤ 2k + 1 < 2(k + 1) = 2γb (G ) by Theorem 2.2 and Corollary 2.4.

(III) Let n ≡ 2 (mod 3). We write n = 3k + 2, where k ≥ 0. Note that γb (G ) = k + 1 by Corollary 2.4. If k = 0, then γb (G σ ) = 2 = 2γb (G ) (for an arbitrary bijection σ ) by (a) of Theorems 2.3 and 4.4. So, let k ≥ 1. First, let σ (u 3 ) = w 3b+1 for some b ∈ {0, 1, . . . , k}. If f : V (G σ ) → {0, 1, . . . , diam(G σ )} is a function defined by

⎧ ⎨ 2 if v = u 3 , −1 { w 3 j +2 }) ∪ (∪kj=b+1 { w 3 j +1 }), f ( v ) = 1 if v ∈ (∪ki=2 {u 3i +1 }) ∪ (∪bj = 0 ⎩ 0 otherwise,

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2 + (k − 1) + b + (k − b) = 2k + 1; thus γb (G σ ) ≤ 2k + 1 < 2(k + 1) = 2γb (G ). Second, let σ (u 3 ) = w 3b+2 for some b ∈ {0, 1, . . . , k}. By fixing the labeling of vertices of G 1 and by relabeling the vertices of G 2 in the reverse order, each permutation graph in this case is isomorphic to some permutation graph considered in the first case. Third, let σ (u 3 ) = w 3b for some b ∈ {1, . . . , k}. If g : V (G σ ) → {0, 1, . . . , diam(G σ )} is a function defined by

⎧ ⎨ 3 if v = u 3 , −2 { w 3 j +2 }) ∪ (∪kj=b+1 { w 3 j +1 }), g ( v ) = 1 if v ∈ (∪ki=2 {u 3i +1 }) ∪ (∪bj = 0 ⎩ 0 otherwise,

then g is a dominating broadcast of G σ with c g ( V (G σ )) = 3 + (k − 1) + (b − 1) + (k − b) = 2k + 1, and hence 2k + 1 < 2(k + 1) = 2γb (G ). Thus, γb (G σ ) = 2γb (G ) if and only if G = P 2 and σ is an arbitrary bijection.

γb (G σ ) ≤

(IV) Let n ≡ 0 (mod 3). We write n = 3k, where k ≥ 1. Note that γb (G ) = k by Corollary 2.4. If k = 1, then γb (G σ ) = 2 = 2γb (G ) (for an arbitrary bijection σ ) by (a) of Theorems 2.3 and 4.4. If k = 2, then γb (G σ ) = 2γb (G ) if and only if Gσ ∼ = P 6  P 2 by Lemma 5.3. So, let k ≥ 3; we show that γb (G σ ) < 2γb (G ) in this case. Case 1: There exist two distinct a, b ∈ {0, 1, . . . k − 1} such that d G 2 (σ (u 3a+2 ), σ (u 3b+2 )) = 1. Subcase 1.1: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x , w 3x+1 } for some x ∈ {1, 2, . . . , k − 1}. Let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function be defined by

⎧ ⎨ 3 if v = w 3x , −1 1 {u 3i +2 }) − {u 3a+2 , u 3b+2 }] ∪ [(∪kj − f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) − { w 3x−1 , w 3x+2 }], ⎩ 0 otherwise.

Then f is a dominating broadcast of G σ with c f ( V (G σ )) = 3 +(k − 2) +(k − 2) = 2k − 1; thus γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Subcase 1.2: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x+1 , w 3x+2 } for some x ∈ {0, 1, . . . , k − 1}. First, let x = 0. If f : V (G σ ) → {0, 1, . . . , diam(G σ )} is a function defined by

⎧ ⎨ 3 if v = w 3x+1 , −1 1 {u 3i +2 }) − {u 3a+2 , u 3b+2 }] ∪ [(∪kj − f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) − { w 3x−1 , w 3x+2 }], ⎩ 0 otherwise,

(1)

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2k − 1; thus γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Second, let x = 0. If there exists c ∈ {0, 1, . . . , k − 1} − {a, b} such that σ (u 3c +2 ) = w 3z for some z ∈ {1, 2, . . . , k}, let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

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 f (v ) =

11

−1 1 k −1 1 if v ∈ (∪ki = {u 3i +2 }) ∪ (∪zj− 0 =1 { w 3 j +1 }) ∪ (∪ j = z { w 3 j +2 }), 0 otherwise;

then f is a dominating broadcast of G σ with c f ( V (G σ )) = k + (k − 1) = 2k − 1. If there exists c ∈ {0, 1, . . . , k − 1} − {a, b} such that σ (u 3c +2 ) = w 3z+1 or σ (u 3c +2 ) = w 3z+2 for some z ∈ {1, 2, . . . , k − 1}, let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 2 if v = σ (u 3c+2 ), −1 1 k −1 {u 3i +2 }) − {u 3c+2 }] ∪ (∪zj− g ( v ) = 1 if v ∈ [(∪ki = 0 =1 { w 3 j +1 }) ∪ (∪ j = z+1 { w 3 j +2 }), ⎩ 0 otherwise;

then g is a dominating broadcast of G σ with c g ( V (G σ )) = 2 + (k − 1) + ( z − 1) + (k − 1 − z) = 2k − 1. In each case, γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Subcase 1.3: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x+2 , w 3x+3 } for some x ∈ {0, 1, . . . , k − 1}. By fixing the labeling of G 1 and by relabeling the vertices of G 2 in the reverse order, each permutation graph in this case is isomorphic to some permutation graph in Subcase 1.2. Case 2: For every pair α , β ∈ {0, 1, . . . , k − 1}, d G 2 (σ (u 3α +2 ), σ (u 3β+2 )) ≥ 2, and there exist two distinct a, b ∈ {0, 1, . . . k − 1} such that d G 2 (σ (u 3a+2 ), σ (u 3b+2 )) = 2. Subcase 2.1: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x , w 3x+2 } for some x ∈ {1, 2, . . . , k − 1}. If f : V (G σ ) → {0, 1, . . . , diam(G σ )} is a function defined as in (1), then f is a dominating broadcast of G σ , and thus γb (G σ ) ≤ c f ( V (G σ )) = 2k − 1 < 2k = 2γb (G ). Subcase 2.2: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x+1 , w 3x+3 } for some x ∈ {0, 1, . . . , k − 1}. Keeping in mind of the assumption that d G 2 (σ (u 3α +2 ), σ (u 3β+2 )) ≥ 2 for every pair α , β ∈ {0, 1, . . . , k − 1}, we consider six possibilities. First, (1) suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x+1 , w 3x+3 , w 3x+5 } for some x ∈ {0, 1, . . . , k − 2} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−1 , w 3x+1 , w 3x+3 } for some x ∈ {1, 2, . . . , k − 1}, say the former, without loss of generality. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 4 if v = w 3x+3 , −1 1 {u 3i +2 }) − {u 3a+2 , u 3b+2 , u 3c+2 }] ∪ [(∪kj − f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) − { w 3x+2 , w 3x+5 }], ⎩ 0 otherwise,

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2k − 1, and hence γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Second, (2) suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x+1 , w 3x+3 , w 3x+6 } for some x ∈ {0, 1, . . . , k − 2} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−2 , w 3x+1 , w 3x+3 } for some x ∈ {1, 2, . . . , k − 1}, say the former, without loss of generality. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 3 if v = w 3x+2 , −1 1 {u 3i +2 }) − {u 3a+2 , u 3b+2 }] ∪ [(∪kj− f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) − { w 3x+2 , w 3x+5 }], ⎩ 0 otherwise,

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2k − 1, and hence γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Third, (3) suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x+1 , w 3x+3 , w 3x+7 } for some x ∈ {0, 1, . . . , k − 3} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−3 , w 3x+1 , w 3x+3 } for some x ∈ {2, 3, . . . , k − 1}, say the former, without loss of generality. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ 3 if v = w 3x+2 , ⎪ ⎪ ⎨ 2 if v = w 3x+7 , f (v ) = k −1 1 1 if v ∈ [(∪ {u 3i +2 }) − {u 3a+2 , u 3b+2 , u 3c+2 }] ∪ [(∪kj − ⎪ =0 { w 3 j +2 }) − { w 3x+2 , w 3x+5 , w 3x+8 }], i =0 ⎪ ⎩ 0 otherwise,

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2k − 1, and hence γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Fourth, (4) suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x+1 , w 3x+3 , w 3x+8 } for some x ∈ {0, 1, . . . , k − 3} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−4 , w 3x+1 , w 3x+3 } for some x ∈ {2, 3, . . . , k − 1}, say the former, without loss of generality. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ 3 if v = w 3x+2 , ⎪ ⎪ ⎨ 2 if v = w 3x+8 , f (v ) = k −1 1 1 if v ∈ [(∪ {u 3i +2 }) − {u 3a+2 , u 3b+2 , u 3c+2 }] ∪ [(∪kj − ⎪ =0 { w 3 j +2 }) − { w 3x+2 , w 3x+5 , w 3x+8 }], i =0 ⎪ ⎩ 0 otherwise,

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12

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2k − 1, and hence γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Fifth, (5) suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x+1 , w 3x+3 , w 3x+9 } for some x ∈ {0, 1, . . . , k − 3} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−5 , w 3x+1 , w 3x+3 } for some x ∈ {2, 3, . . . , k − 1}, say the former, without loss of generality. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 2 if v = w 3x+3 , −1 1 k −1 {u 3i +2 }) − {σ −1 ( w 3x+3 )}] ∪ [(∪xj− f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) ∪ { w 3x+7 } ∪ (∪ j =x+3 { w 3 j +2 })], ⎩ 0 otherwise;

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2k − 1, and hence γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Sixth, assume that there are no three distinct a, b, c ∈ {0, 1, . . . , k − 1} satisfying any of the conditions (1)–(5). This, combined together with the assumptions of Case 2 and Subcase 2.2, implies that k = 4 and ∪4i =1 {σ (u 3i +2 )} = { w 1 , w 3 , w 10 , w 12 }; then the function f : V (G σ ) → {0, 1, . . . , diam(G σ )} defined by

⎧ ⎨ 3 if v ∈ { w 2 , w 11 }, f ( v ) = 1 if v = w 6 , ⎩ 0

otherwise

is a dominating broadcast of G σ with γb (G σ ) ≤ c f ( V (G σ )) = 7 < 8 = 2γb (G ). Subcase 2.3: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x−1 , w 3x+1 } for some x ∈ {1, 2, . . . , k − 1}. By fixing the labeling of G 1 and by relabeling the vertices of G 2 in the reverse order, each permutation graph in this case is isomorphic to some permutation graph in Subcase 2.1. Case 3: For every pair α , β ∈ {0, 1, . . . k − 1}, d G 2 (σ (u 3α +2 ), σ (u 3β+2 )) ≥ 3, and there exist two distinct a, b ∈ {0, 1, . . . k − 1} such that d G 2 (σ (u 3a+2 ), σ (u 3b+2 )) = 3. Subcase 3.1: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x , w 3x+3 } for some x ∈ {1, 2, . . . , k − 1}. Let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 2 if v = w 3x , −1 1 {u 3i +2 }) − {σ −1 ( w 3x )}] ∪ [(∪kj− f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) − { w 3x−1 , w 3x+2 }], ⎩ 0 otherwise.

Then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2 +(k − 1) +(k − 2) = 2k − 1; thus γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Subcase 3.2: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x−2 , w 3x+1 } for some x ∈ {1, 2, . . . , k − 1}. By fixing the labeling of G 1 and by relabeling the vertices of G 2 in the reverse order, each permutation graph in this case is isomorphic to some permutation graph in Subcase 3.1. Subcase 3.3: {σ (u 3a+2 ), σ (u 3b+2 )} = { w 3x−1 , w 3x+2 } for some x ∈ {1, 2, . . . , k − 1}. Note that 3 ≤ d G 2 (σ (u 3α +2 ), σ (u 3β+2 )) ≤ 4 for every pair α , β ∈ {0, 1, . . . , k − 1} by the assumptions of Case 3 and Subcase 3.3. First, suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x−1 , w 3x+2 , w 3x+5 } for some x ∈ {1, 2, . . . , k − 2} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−4 , w 3x−1 , w 3x+2 } for some x ∈ {2, 3, . . . , k − 1}, say the former, without loss of generality. If we let f : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 2 if v ∈ { w 3x−1 , w 3x+5 }, −1 1 {u 3i +2 }) − {σ −1 ( w 3x−1 ), σ −1 ( w 3x+5 )}] ∪ [(∪kj − f ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) − { w 3x−1 , w 3x+2 , w 3x+5 }], ⎩ 0 otherwise,

then f is a dominating broadcast of G σ with c f ( V (G σ )) = 2(2) + (k − 2) + (k − 3) = 2k − 1; thus γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ). Second, suppose that there exist three distinct a, b, c ∈ {0, 1, . . . , k − 1} such that {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c +2 )} = { w 3x−1 , w 3x+2 , w 3x+6 } for some x ∈ {1, 2, . . . , k − 2} or {σ (u 3a+2 ), σ (u 3b+2 ), σ (u 3c+2 )} = { w 3x−5 , w 3x−1 , w 3x+2 } for some x ∈ {2, 3, . . . , k − 1}, say the former, without loss of generality. If we let g : V (G σ ) → {0, 1, . . . , diam(G σ )} be a function defined by

⎧ ⎨ 2 if v = w 3x−1 , −1 2 k −1 {u 3i +2 }) − {σ −1 ( w 3x−1 )}] ∪ [(∪xj− g ( v ) = 1 if v ∈ [(∪ki = 0 =0 { w 3 j +2 }) ∪ { w 3x+4 } ∪ (∪ j =x+2 { w 3 j +2 })], ⎩ 0 otherwise,

then g is a dominating broadcast of G σ with c g ( V (G σ )) = 2k − 1; thus

γb (G σ ) ≤ 2k − 1 < 2k = 2γb (G ).

Case 4: For every pair α , β ∈ {0, 1, . . . k − 1}, d G 2 (σ (u 3α +2 ), σ (u 3β+2 )) ≥ 4. In order for k distinct vertices to be placed at distance at least four apart on a 3k-path, we have 1 + 4(k − 1) ≤ 3k, which implies k ≤ 3. Since k ≥ 3, this case can occur

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only when k = 3 and {σ (u 2 ), σ (u 5 ), σ (u 8 )} = { w 1 , w 5 , w 9 }. Then S = {u 2 , u 5 , u 8 , w 3 , w 7 } forms a dominating set of G σ with | S | = 5; thus, γb (G σ ) ≤ 5 < 2γb (G ) by Theorem 2.2. 2 Next, we consider permutation graphs on complete multipartite graphs. Theorem 5.5. For k ≥ 2, let G = K a1 ,a2 ,...,ak be a complete k-partite graph of order at least three, and let s be the number of partite sets of G consisting of exactly one element. Let G 1 and G 2 be two disjoint copies of G, and let σ : V (G 1 ) → V (G 2 ) be a bijection. Let V (G 1 ) be partitioned into k-partite sets V 1 , V 2 , . . . , V k , and let V (G 2 ) be partitioned into k-partite sets V 1 , V 2 , . . . , V k , where | V i | = ai = | V i | for i ∈ {1, 2, . . . , k}. Then γb (G σ ) ∈ {2, 3}. Moreover, γb (G σ ) = 2 if and only if one of the followings holds: (i) s = 0 and σ is an arbitrary bijection, or (ii) s = 0 and |σ ( V i ) ∩ V j | = 1 for some i , j ∈ {1, 2, . . . , k}, or

(iii) s = 0 and there exist V x , V y such that | V x | = 2 = | V y | with σ ( V x ) = V y . Proof. Since rad(G σ ) ≤ 3, γb (G σ ) ∈ {2, 3} by Theorem 2.2 and Proposition 3.1. Next, we characterize G σ satisfying γb (G σ ) = 2. Case 1: s = 0. In this case, (G ) = | V (G )| − 1; thus γb (G σ ) = 2 = 2γb (G ) by (a) of Theorems 2.3 and 4.4. Case 2: s = 0. In this case, ai ≥ 2 for each i ∈ {1, 2, . . . , k}; thus G σ fails to satisfy (a) of Theorem 4.4. First, let |σ ( V i ) ∩ V j | = 1 for some i , j ∈ {1, 2, . . . , k}. Then rad(G σ ) = 2, and thus γb (G σ ) = 2 by Theorem 2.3. Second, let |σ ( V i ) ∩ V j | ≥ 2 for each i , j ∈ {1, 2, . . . , k}; then rad(G σ ) = 3 and G σ fails to satisfy (c) of Theorem 4.4. If

there exist V i and V j with | V i | = 2 = | V j | such that σ ( V i ) = V j , say V i = {u , u  } and V j = { w , w  } with σ (u ) = w, then S = {u , w } forms a dominating set of G σ with | S | = 2; thus γ (G σ ) = 2 and hence γb (G σ ) = 2 by Theorem 2.3. If there is no i , j ∈ {1, 2, . . . , k} satisfying | V i | = 2 = | V j | and σ ( V i ) = V j , then G σ fails to satisfy (b) of Theorem 4.4; thus γb (G σ ) = 2, and hence γb (G σ ) = 3 by Proposition 3.1 and Theorem 4.4. 2 Acknowledgements The author thanks the anonymous referees for their valuable comments and suggestions. References [1] C. Balbuena, X. Marcote, P. García-Vázquez, On restricted connectivities of permutation graphs, Networks 45 (2005) 113–118. [2] J.R.S. Blair, P. Heggernes, S. Horton, F. Manne, Broadcast domination algorithms for interval graphs, series-parallel graphs, and trees, Congr. Numer. 169 (2004) 55–77. [3] G. Chartrand, F. Harary, Planar permutation graphs, Ann. Inst. Henri Poincaré (Sect. B) 3 (1967) 433–438. [4] J. Dunbar, D. Erwin, T. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, Broadcasts in graphs, Discrete Appl. Math. 154 (2006) 59–75. [5] L. Eroh, R. Gera, C.X. Kang, C.E. Larson, E. Yi, Domination in functigraphs, Discuss. Math., Graph Theory 32 (2) (2012) 299–319. [6] D.J. Erwin, Cost Domination in Graphs, Dissertation, Western Michigan University, 2001. [7] D.J. Erwin, Dominating broadcasts in graphs, Bull. Inst. Comb. Appl. 42 (2004) 89–105. [8] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York, 1979. [9] W. Gu, On diameter of permutation graphs, Networks 33 (1999) 161–166. [10] W. Gu, K. Wash, Bounds on the domination number of permutation graphs, J. Interconnect. Netw. 10 (3) (2009) 205–217. [11] P. Heggernes, D. Lokshtanov, Optimal broadcast domination in polynomial time, Discrete Math. 306 (2006) 3267–3280. [12] F. Hurtado, M. Mora, E. Rivera-Campo, R. Zuazua, Distance 2-domination in prisms of graphs, Discuss. Math., Graph Theory 37 (2017) 383–397. [13] C.L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. [14] A. Meir, J.W. Moon, Relations between packing and covering number of a tree, Pac. J. Math. 61 (1975) 225–233.