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Mutual transferability for (F,B,R) -domination on strongly chordal graphs and cactus graphs
Discrete Applied Mathematics 259 (2019) 41–52
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Mutual transferability for (F , B, R)-domination on strongly chordal graphs and cactus graphs✩ ∗
Kuan-Ting Chu a , Wu-Hsiung Lin b , , Chiuyuan Chen b a b
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
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Article history: Received 30 April 2018 Received in revised form 22 October 2018 Accepted 28 December 2018 Available online 23 January 2019 Keywords: Domination Stability Transferability Strongly chordal graphs Cactus graphs
a b s t r a c t This paper studies a variation of domination in graphs called (F , B, R)-domination. Let G = (V , E) be a graph and V be the disjoint union of F , B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. An (F , B, R)dominating set of G is a subset D ⊆ V such that R ⊆ D and each vertex in B − D is adjacent to some vertex in D. An (F , B, R)-2-stable set of G is a subset S ⊆ B such that S ∩ N(R) = ∅ and every two distinct vertices x and y in S have distance d(x, y) > 2. We prove that if G is strongly chordal, then αF ,B,R,2 (G) = γF ,B,R (G)−|R|, where γF ,B,R (G) is the minimum cardinality of an (F , B, R)-dominating set of G and αF ,B,R,2 (G) is the maximum cardinality of an (F , B, R)∗
2-stable set of G. Let D1 → D2 denote D1 being transferable to D2 . We prove that if G is a connected strongly chordal graph in which D1 and D2 are two (F , B, R)-dominating sets ∗ with |D1 | = |D2 |, then D1 → D2 . We also prove that if G is a cactus graph in which D1 and D2 ∗ are two (F , B, R)-dominating sets with |D1 | = |D2 |, then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, where ∪{1 · extra} means adding one extra element. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Domination and its variations in graphs have been widely studied in the literature and they have numerous applications in computer networks. Especially, they can be used in server allocation to ensure the Quality of Service (QoS) of a network. Let G = (V , E) be a graph. The neighborhood N(v ) of a vertex v is the set to v ; the closed neighborhood ⋃of all vertices adjacent⋃ N [v] of v is N [v] = N(v ) ∪ {v}. For any subset S ⊆ V , define N(S) = v∈S N(v ) and N [S ] = v∈S N [v]. A vertex v is said to dominate all vertices in N [v]. A subset D ⊆ V is a dominating set of G if every vertex ⋃ in V − D is adjacent to at least one vertex in D. Chang [4] pointed out that this is equivalent to N [v] ∩ D ̸ = ∅ for all v ∈ V or u∈D N [u] = V . A subset S ⊆ V is a 2-stable set of G if every two distinct vertices x and y in S have d(x, y) > 2. The domination number γ (G) of G is the minimum cardinality of a dominating set of G. The 2-stability number α2 (G) of G is the maximum cardinality of a 2-stable set of G. It is easy to see that any graph G has (Weak Duality Inequality for Domination): α2 (G) ≤ γ (G). Chang [4] used an algorithmic approach to show that any tree T has (Strong Duality Equality for Domination): α2 (T ) = γ (T ). Such a primal–dual approach was used by Farber [13] and Kolen [17] for weighted domination on strongly chordal graphs. Some definitions and notations that follow the paper of Fujita [14] must be introduced. Let D1 and D2 be two dominating sets of G. We say that D1 is single-step transferable to D2 , denoted as D1 → D2 , if there exist u ∈ D1 and v ∈ D2 such that ✩ This research was partially supported by the Ministry of Science and Technology of Taiwan under grants MOST-106-2115-M-009-005 and MOST-1062115-M-009-007. ∗ Corresponding author. E-mail address: [email protected] (W.-H. Lin). https://doi.org/10.1016/j.dam.2018.12.034 0166-218X/© 2019 Elsevier B.V. All rights reserved.
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∗
Fig. 1. Let D1 = {a, b, e}, D2 = {a, c , f }, and D3 = {a, c , e}. Then, D1 → D3 , D3 → D2 , and D1 → D2 .
∗
uv ∈ E and D1 − {u} = D2 − {v}. We write D1 → D2 if D1 can be transferred to D2 through a sequence of single-step transfers ∗ (also called a sequence of safe moves in [14]). By definition, D1 → D2 implies that D2 → D1 and |D1 | = |D2 |; also, D1 → D2 ∗ implies that D2 → D1 and |D1 | = |D2 |. Fig. 1 gives an illustration. For technical reasons, [14] assumes that a dominating set of G is a dominating multiset (which is like a set except that its members need not be distinct); in this paper, we make the same assumption. Reference [14] studies the following server allocation problem: Given two dominating sets D1 and D2 of a graph G, can we transfer D1 to D2 through a sequence of safe moves? The main results of [14] are as follows. ∗
• If T is a tree with n vertices in which D1 and D2 are two dominating sets with |D1 | = |D2 | ≥ ⌊ 2n ⌋, then D1 → D2 . • If G is a Hamiltonian graph with n vertices in which D1 and D2 are two dominating sets with |D1 | = |D2 | ≥ ⌈ n+3 1 ⌉, then ∗ D1 → D2 . The purpose of this paper is to study (F , B, R)-domination (also called mixed domination) for strongly chordal graphs and cactus graphs. Let G = (V , E) be a graph and suppose V is the disjoint union of three sets F , B, and R, where F means ‘‘Free’’, consisting of free vertices, B means ‘‘Bound’’, consisting of bound vertices, and R means ‘‘Required’’, consisting of required vertices. An (F , B, R)-dominating set of G is a subset D ⊆ V such that R ⊆ D and each vertex in B − D is adjacent to some vertex in D; free vertices need not be dominated by D but may be included in D to dominate vertices. (See [8,16].) Notice that ordinary domination is the same as (F , B, R)-domination when B = V (G), F = R = ∅. An (F , B, R)-2-stable set of G is a subset S ⊆ B such that S ∩ N(R) = ∅ and every two distinct vertices x and y in S have d(x, y) > 2. The (F , B, R)-domination number γF ,B,R (G) of G is the minimum cardinality of an (F , B, R)-dominating set of G. The (F , B, R)-2-stability number αF ,B,R,2 (G) of G is the maximum cardinality of an (F , B, R)-2-stable set of G. By replacing ‘‘dominating set’’ with ‘‘(F , B, R)-dominating set’’ ∗ in the definitions of D1 → D2 and D1 → D2 , mutual transferability among (F , B, R)-dominating sets D1 and D2 of G can be defined and we omit the details. We have one remark for (F , B, R)-dominating sets. Up to now, finding a minimum (F , B, R)-dominating set is regarded as an approach (a labeling method) for deriving a minimum dominating set. The final goal is a minimum dominating set, not a minimum (F , B, R)-dominating set. However, as can be seen from Fig. 2 and the following scenario, a minimum (F , B, R)dominating set is more flexible than a minimum dominating set when we want to model a real-world problem.
• Some of the cities in Taiwan are the most important; for example, Taipei, Hsinchu, and Kaohsiung. We need to allocate a server for each of them and therefore mark each of them as required.
• Some of the cities in Taiwan are important but not the most important; for example, Taoyuan, Taichung, Tainan, and so on. We do not need to allocate a server for each of them but we need to ensure that each of them has a server within its closed neighborhood. We thus mark each of them as bound. • Some of the cities in Taiwan are not so important and we do not even have to make each of them have a server within its closed neighborhood. We therefore mark each of them as free. It is easy to see the following inequality. Weak Duality Inequality for (F , B, R) -Domination: αF ,B,R,2 (G) ≤ γF ,B,R (G) − |R| for any graph G. The inequality can be strict, as shown by the 5-cycle C5 (with B being the set of all vertices and F = R = ∅) that αF ,B,R,2 (C5 ) = 1 < 2 = γF ,B,R (C5 ) − |R|. Main results of this paper are:
• Any strongly chordal graph G has (Strong Duality Equality for (F , B, R)-Domination): αF ,B,R,2 (G) = γF ,B,R (G) − |R|. • If G is a connected strongly chordal graph in which D1 and D2 are two (F , B, R)-dominating sets with |D1 | = |D2 |, then ∗ D1 → D2 . ∗ • If G is a cactus graph in which D1 and D2 are two (F , B, R)-dominating sets with |D1 | = |D2 |, then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. Definitions of strongly chordal graphs, cactus graphs, and ∪{1 · extra} will be given later. It is well known that strongly chordal graphs is a subclass of chordal graphs. On the other hand, trees, block graphs, interval graphs, and directed path
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Fig. 2. Map of Taiwan.
graphs are subclasses of strongly chordal graphs. See [2,9] for definitions of these graphs. Containment relations between these graphs can be shown below: tree interval graph
⊂ ⊂
block graph directed path graph
⊂ ⊂
strongly chordal graph strongly chordal graph
⊂ ⊂
chordal graph chordal graph
It follows that our second main result applies to trees, block graphs, interval graphs, and directed path graphs. Thus it improves the first main result of [14], which only applies to trees and dominating sets of cardinality ≥ ⌊ 2n ⌋. Every cycle C3r +6 , r ≥ 0, is a cactus graph. However, no two distinct minimum dominating sets of C3r +6 are mutually transferable; therefore at least one extra element has to be added. Take the C6 shown in Fig. 1 as an example. Both {b, e} and {c , f } are minimum dominating sets of C6 but they are not mutually transferable. After the extra element a is added to {b, e} and also ∗ to {c , f }, we have {b, e} ∪ {a} → {c , f } ∪ {a}. Our third main result (which adds only one extra element) is therefore the best possible. This paper is organized as follows. Section 2 gives preliminaries. Section 3 establishes the strong duality equality for (F , B, R)-domination on strongly chordal graphs. Sections 4 and 5 prove mutual transferability for (F , B, R)-domination on connected strongly chordal graphs and on cactus graphs, respectively. Concluding remarks are drawn in Section 6. 2. Preliminaries This paper considers undirected graphs G = (V , E) without loops or multiple edges. We now define strongly chordal graphs. Suppose G = (V , E) is a graph and v1 , v2 , . . . , vn is an ordering of V (G). Set Ni [vj ] = N [vj ] ∩ {vi , vi+1 , . . . , vn } for easy writing. Two definitions are given. Definition 1. A strong elimination ordering of a graph G = (V , E) is an ordering v1 , v2 , . . . , vn of V such that for each i, j, and k, if i ≤ j ≤ k and vj , vk ∈ N [vi ], then Ni [vj ] ⊆ Ni [vk ]. Definition 2. A graph is strongly chordal if it admits a strong elimination ordering. For convenience, throughout this paper, vi∗ is used to denote the largest-indexed vertex in N [vi ]. The class of strongly chordal graphs was introduced by Farber in [11]; see also [12,13]. Chang and Nemhauser [3,5] independently discovered this class of graphs. Reference [12] provides a polynomial-time algorithm to recognize strongly chordal graphs and to construct strong elimination orderings; see also [1,19–21]. Actually, several versions of strong elimination orderings have been proposed (see [6,12,13,15,22]) and can be proven to be equivalent. In this paper, we assume that a strong elimination ordering is always available when the given graph is strongly chordal. We now are ready to define the notion ∪{1 · extra}. To do this, we actually define the notion D ∪ {ℓ · extra}. Intuitively, when an (F , B, R)-dominating set of G contains more than γF ,B,R (G) elements, it may contain elements whose deletion will not affect the (F , B, R)-domination and we call these elements extra elements. More precisely, suppose D is already an (F , B, R)dominating set of G and {s1 , s2 , . . . , sℓ } is a subset of V (G). Then, D ∪ {s1 , s2 , . . . , sℓ } is an (F , B, R)-dominating set of G and s1 , s2 , . . . , sℓ are extra elements. When the names of s1 , s2 , . . . , sℓ are unimportant, we simply write D ∪ {s1 , s2 , . . . , sℓ } as D ∪ {ℓ · extra}. By setting ℓ = 1 into D ∪ {ℓ · extra}, we have D ∪ {1 · extra} and the meaning of the notion ∪{1 · extra} is now ∗ clear. Notice that when D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, a sequence of single-step transfers from D1 to D2 can contain an (F , B, R)-dominating set of G that has no extra element; D3 in Fig. 1 is an example. Before closing this section, we briefly describe the background of the server allocation problem studied by Fujita [14]. The purpose of this problem is to guarantee a continuous service to all users of a network by a collection of mobile servers distributed over the network. Let D be the (multi)set of nodes that have mobile servers. Then, all users can receive the service
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of the network if D is a dominating (multi)set of the network’s corresponding graph G. Since mobile servers can move, to guarantee a continuous service to all users, the change of servers’ locations must obey three rules:
• A mobile server can move from a node to any adjacent node in a single step. • In each step, at most one mobile server can move. • Each move does not leave any node not dominated after the move. Notice that during the move of a mobile server, some nodes might be not dominated. Also notice that for (F , B, R)-domination, only bound vertices need to be dominated. A move of a mobile server is said to be safe if it obeys the above rules; this definition is crucial for the rest of this paper. Based on the above rules, [14] studies the mutual transferability among ∗ dominating sets D1 and D2 of G and asks if D1 → D2 . 3. Strong duality equality for (F , B, R)-domination on strongly chordal graphs The study of domination in strongly chordal graphs by Farber [11,13], Chang and Nemhauser [5] and Kolen [17] also leads to the strong duality equality γ (G) = α2 (G). On the other hand, the approaches of labeling algorithms for (F , B, R)domination (for example, Cockayne et al. [8] for trees, Laskar et al. [18] for total domination, Hedetniemi et al. [16] for cactus graphs, Chang [4] for strongly chordal graphs) are somehow ad hoc. In this section we give a primal–dual approach for the (F , B, R)-domination on strongly chordal graphs, with the strong duality equality as a by product. Using the same spirit, we give a mutual transferability in the next section. In particular, we show that αF ,B,R,2 (G) = γF ,B,R (G) − |R| for any strongly chordal graph G. We do this by proposing an algorithm, called Algorithm 1, to find both a minimum (F , B, R)-dominating set D∗ and a maximum (F , B, R)-2-stable set S ∗ of G. Here we briefly describe the idea behind Algorithm 1. Suppose v1 , v2 , . . . , vn is a strong elimination ordering of G and vi is a bound vertex. Then, at least one vertex in N [vi ] has to be put into D∗ to dominate vi . Recall that we denote by vi∗ the largest-indexed vertex in N [vi ]. Therefore suppose N [vi ] = {vi1 , vi2 , . . . , viℓ }, where i1 < i2 < · · · < iℓ . Then, by definition, vi∗ = viℓ . Since Ni [vi1 ] ⊆ Ni [vi2 ] ⊆ · · · ⊆ Ni [vi∗ ], Algorithm 1 puts vi∗ into D∗ . Besides, Algorithm 1 puts vi into S ∗ . Notice that Farber [13] has proposed an algorithm to find a minimum weighted dominating set for any strongly chordal graph. Farber’s algorithm can find a minimum dominating set, but not a minimum (F , B, R)-dominating set. Algorithm 1 Input: A strongly chordal graph G = (V , E) with a strong elimination ordering v1 , v2 , . . . , vn and a partition F , B, R of V , where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. Output: A minimum (F , B, R)-dominating set D∗ and a maximum (F , B, R)-2-stable set S ∗ of G. ∗ 1: D ← R; ∗ 2: S ← ∅; 3: relabel all the bound vertices in N(R) as free; 4: k ← 0; 5: for i = 1 to n do 6: if (vi is bound) then 7: k ← k + 1; 8: bk ← v i ; 9: rk ← vi∗ ; //b1 , b2 , . . . , bk and r1 , r2 , . . . , rk are used in proving correctness and mutually transferability 10: relabel vi∗ as required; 11: relabel all the bound vertices in N(vi∗ ) as free; 12: D∗ ← D∗ ∪ {vi∗ }; 13: S ∗ ← S ∗ ∪ {vi }; 14: end if 15: end for Fig. 3 shows an example of Algorithm 1. Two claims are made. Claim 3.1. N [R] ∩ {b1 , b2 , . . . , bk } = ∅ and R ∩ {r1 , r2 , . . . , rk } = ∅. Proof. During the execution of line 3 of Algorithm 1, all the bound vertices in N(R) are relabeled as free. Since b1 , b2 , . . . , bk are identified during the execution of line 8 of Algorithm 1, none of b1 , b2 , . . . , bk is a bound vertex in N(R). Hence N [R] ∩ {b1 , b2 , . . . , bk } = ∅. Thus R cannot dominate b1 , b2 , . . . , bk . Therefore R ∩ {r1 , r2 , . . . , rk } = ∅. □ Claim 3.2. d(bi , bj ) > 2 for any 1 ≤ i < j ≤ k. Proof. Notice that ri is the largest-indexed vertex in N [bi ]. We first prove that d(bi , bj ) = 1 is impossible. Suppose d(bi , bj ) = 1. Then, bj ∈ N [bi ]. By the property of a strongly chordal graph, bj ∈ N [ri ] must occur. Thus once
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Fig. 3. (a) A strongly chordal graph with F = ∅, B = {v1 , v2 , . . . , v9 }, R = ∅, and strong elimination ordering v1 , v2 , . . . , v9 . (b) Identifying b1 and r1 . (c) Identifying b2 and r2 . (d) Identifying b3 and r3 . Output will be D∗ = {r1 , r2 , r3 } = {v3 , v6 , v9 } and S ∗ = {b1 , b2 , b3 } = {v1 , v5 , v8 }.
Algorithm 1 identifies ri (in order to dominate bi ), it relabels the vertex that corresponds to bj as free (see line 11 of Algorithm 1). Hence the vertex that corresponds to bj is no longer a bound vertex; so d(bi , bj ) = 1 is impossible. We now prove that d(bi , bj ) = 2 is also impossible. Suppose d(bi , bj ) = 2. Then, bi and bj have a common neighbor vt0 . Let vt1 and vt2 be the vertices corresponding to bi and bj , respectively. It is clear that t1 < t2 . There are two cases. Case 1: t0 < t1 . Then, since vt1 , vt2 ∈ N [vt0 ], by the property of the strong elimination ordering, Nt0 [vt1 ] ⊆ Nt0 [vt2 ] must occur. Since bi ∈ Nt0 [vt1 ] and Nt0 [vt2 ] ⊆ N [bj ], it follows that bi ∈ N [bj ], contradicting with the assumption that d(bi , bj ) = 2. Case 2: t1 < t0 . Then, either vt0 and ri are the same vertices or they are different. In the former case, vt2 ∈ N [ri ] is assured. In the latter case, vt0 must proceed ri in the strong elimination ordering. Then, since vt0 , ri ∈ N [vt1 ], by the property of the strong elimination ordering, we have Nt1 [vt0 ] ⊆ Nt1 [ri ]. Since vt2 ∈ Nt1 [vt0 ] and Nt1 [ri ] ⊆ N [ri ], it follows that vt2 ∈ N [ri ] is also assured. Hence, once Algorithm 1 identifies ri (in order to dominate bi ), it relabels vt2 as free, contradicting with the assumption that bj is bound. We thus have this claim. □ The correctness of Algorithm 1 and the strong duality equality for (F , B, R)-domination can now be proved. Theorem 3.3. By using the adjacency-list representation of graphs, Algorithm 1 gives a minimum (F , B, R)-dominating set D∗ and a maximum (F , B, R)-2-stable set S ∗ of a strongly chordal graph G = (V , E) in O(|V | + |E |) time provided that a strong elimination ordering is given. Moreover, the strong duality equality αF ,B,R,2 (G) = γF ,B,R (G) − |R| holds. Proof. It is clear that D∗ = R ∪ {r1 , r2 , . . . , rk } and S ∗ = {b1 , b2 , . . . , bk }. Since each bi is a bound vertex, S ∗ ⊆ B follows. By Claim 3.1, S ∗ ∩ N [R] = ∅. By Claim 3.2, every two distinct bi and bj in S ∗ have distance d(bi , bj ) > 2. Therefore S ∗ is an (F , B, R)-2-stable set of G. That D∗ is an (F , B, R)-dominating set of G is due to (i)∼(iii) below. (i) Algorithm 1 relabels a bound vertex vi as free only when vi is adjacent to some required vertex (see lines 3 and 11). (ii) Algorithm 1 relabels a bound vertex vi as required only when vi itself is the largest-indexed vertex in N [vi ] or it is the largest-indexed neighbor of a bound vertex proceeding it (see lines 6, 9 and 10). (iii) Algorithm 1 never relabels a required vertex and it includes every required vertex into D∗ (see lines 1, 10, and 12). Since S ∗ is an (F , B, R)-2-stable set and D∗ is an (F , B, R)-dominating set of G, we have |S ∗ | ≤ αF ,B,R,2 (G) and γF ,B,R (G) ≤ |D∗ |. By the weak duality inequality for (F , B, R)-domination, αF ,B,R,2 (G) ≤ γF ,B,R (G) − |R|. By Claim 3.1, R ∩ {r1 , r2 , . . . , rk } = ∅; therefore D∗ − R = {r1 , r2 , . . . , rk } and |D∗ | − |R| = |D∗ − R|. Thus k = |{b1 , b2 , . . . , bk }| = |S ∗ | ≤ αF ,B,R,2 (G) ≤ γF ,B,R (G) − |R| ≤ |D∗ | − |R| = |D∗ − R| = |{r1 , r2 , . . . , rk }| = k. The three inequalities must be equalities. Hence D∗ is a minimum (F , B, R)-dominating set and S ∗ is a maximum (F , B, R)-2stable set of G. Moreover, αF ,B,R,2 (G) = γF ,B,R (G) − |R|.
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We now analyze the time complexity of Algorithm 1. Recall that we use the adjacency-list representation of graphs. The most time-consuming steps in Algorithm 1 are lines 3, 9 and 11; it is not difficult to see that all the other steps can be done in O(|V | + |E |) time. Line 3 takes O(|V | + |E |) time because: |R| ≤ |V | and |N(R)| ≤ 2|E |. Line 9 takes O(|V | + |E |) time because: once the vertices and edges of G have been input, a scan of the adjacency-lists suffices to find vi∗ for every vi and it takes O(|V | + |E |) time. Note that several vertices in G may have the same vi∗ (for example, in Fig. 3, vi∗ = v9 for all i ∈ {7, 8, 9}). Thus we can use a mark array to indicate whether N(vi∗ ) is visited or not and only when N(vi∗ ) is visited for the first time, we relabel all the bound vertices in N(vi∗ ). Based on such an implementation, line 11 can be done in O(|V | + |E |) time. From the above, Algorithm 1 takes O(|V | + |E |) time. □ 4. Mutual transferability for connected strongly chordal graphs The purpose of this section is to prove mutual transferability among (F , B, R)-dominating sets in connected strongly chordal graphs G. The approach of the proof uses the same spirit in Section 3 for the primal–dual method. First a lemma. Lemma 4.1. Let G be a connected strongly chordal graph with a strong elimination ordering v1 , v2 , . . . , vn . If i < n, then the largest-indexed vertex vi∗ in N [vi ] has i∗ > i. Proof. Choose a shortest vi -vn path vi1 vi2 . . . vim with vi1 = vi and vim = vn . Suppose there is some vir with ir < i, say ir is the smallest index with this property. Then 1 < r < m and vir −1 , vir +1 ∈ Nir [vir ], which gives that vir −1 vir +1 ∈ E(G) by the property of the strong elimination ordering. This leads to a shorter vi -vn path, a contradiction. Hence i2 > i and so i∗ > i. □ Theorem 4.2. For any connected strongly chordal graph G, if D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |, ∗ then D1 → D2 . Proof. Choose a strong elimination ordering v1 , v2 , . . . , vn of G. We shall prove a more general statement as follows by a back induction. ∗
(Si ) If D1 and D2 agree on {v1 , v2 , . . . , vi } (i.e., for each 1 ≤ j ≤ i, vj ∈ D1 iff vj ∈ D2 ), then D1 → D2 . Notice that the theorem is just (S0 ). (Sn ) is true as in this case D1 = D2 . Suppose 1 ≤ i ≤ n and (Si ) is true. To prove (Si−1 ) being true, suppose D1 and D2 agree on {v1 , v2 , . . . , vi−1 }. Suppose D1 has c1 copies of vi and D2 has c2 copies of vi . We may assume that c1 > c2 since nothing needs to be done if c1 = c2 and the case c2 > c1 can be proven similarly. Also, c1 > c2 ˆi = B ∩ (N [vi ] − N [D1 − {vi }]) for easy writing. Intuitively, N ˆi denotes the set of private bound neighbors implies i < n. Set N of vi with respect to D1 , i.e., bound vertices that only can be dominated by vi . Define j to be the minimum index such that ˆi , where j may or may not exist. There are two cases. vj ∈ N Case 1: c2 > 0 or j does not exist or j ≥ i. By Lemma 4.1, the largest-indexed vertex vi∗ in N [vi ] has i∗ > i. Moving c1 –c2 copies of vi of D1 to vi∗ results a set D′1 . If c2 > 0, then c2 copies of vi of D1 stay at vi of D′1 , and so D′1 is an (F , B, R)-dominating set ∗
with D1 → D′1 . If j does not exist, then vi has no private bound neighbors. If j ≥ i, then all the private bound neighbors of vi with respect to D1 are in Ni [vi ] ⊆ Ni [vi∗ ]. Also c2 = 0 gives vi ∈ F ∪ B. Therefore D′1 is an (F , B, R)-dominating set ∗
with D1 → D′1 . Case 2: c2 = 0 and j < i. Since c2 = 0 and D1 and D2 agree on {v1 , v2 , . . . , vi−1 }, D2 must dominate vj by using a vertex vj′ . Notice that j′ > i since vj is a private bound neighbor of vi and D1 and D2 agree on {v1 , v2 , . . . , vi−1 }. Moving c1 copies of vi of D1 to vj′ results a set D′1 . All the private bound neighbors of vi with respect to D1 are in Nj [vi ] ⊆ Nj [vj′ ]. Also ∗
c2 = 0 gives vi ∈ F ∪ B. Therefore D′1 is an (F , B, R)-dominating set with D1 → D′1 . ∗
∗
In any case, D′1 and D2 agree on {v1 , v2 , . . . , vi }, by the induction hypothesis, D′1 → D2 and so D1 → D2 . (See Fig. 4 for an illustration of the proof.) □ By letting B = V (G) and F = R = ∅, we have the following corollary, which improves the first main result in [14]. Corollary 4.3. If G is a connected strongly chordal graph in which D1 and D2 are two dominating sets with |D1 | = |D2 |, then ∗ D1 → D2 . 5. Mutual transferability for cactus graphs Before going further, we define cactus graphs and describe the relationship between cactus graphs and trees. A cutvertex in a graph is a vertex whose removal increases the number of connected components. A block in a graph is a maximal connected subgraph of the graph having no cut-vertices. A cactus graph is a connected graph in which each block is either an edge or a cycle. Thus a tree is a cactus graph in which each block is an edge. A cactus graph is sometimes called a cactus tree and any two cycles in it have at most one vertex in common. In [10], Das mentioned that cactus graphs have many applications because these graphs can be used to model physical setting (especially in telecommunications and local area
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∗
Fig. 4. The process of D1 → D2 , where D1 = {v2 , v4 , v7 , v7 , v8 } and D2 = {v3 , v6 , v7 , v9 , v9 }, in the connected strongly chordal graph shown in Fig. 3(a), which has F = ∅, B = {v1 , v2 , . . . , v9 }, R = ∅, and strong elimination ordering v1 , v2 , . . . , v9 . (a) (S1 ) H⇒ (S2 ). (b) (S3 ) H⇒ (S4 ). (c) (S4 ) H⇒ (S5 ). (d) ∗
(S6 ) H⇒ (S7 ). (e) (S7 ) H⇒ (S8 ). (f) (S9 ) holds; the process D1 → D2 now completes.
networks) where a tree would be inappropriate. In [16], Hedetniemi et al. proposed a linear-time algorithm, called MCACTUS, for finding a minimum (F , B, R)-dominating set in a cactus graph, which in turn uses an algorithm called MCYCLE to find a minimum (F , B, R)-dominating set in a cycle or an edge. In what follows, let G = (V , E) be a cactus graph and V be the disjoint union of F , B, and R. A block of G is called an end-block if it contains exactly one cut-vertex. We call an end-block of G a cycle end-block if it is a cycle. The rest of this section is organized as follows. Section 5.1 deals with the case that G is exactly a cycle. Section 5.2 deals with the case that G has a cycle end-block. Section 5.3 handles the general case. 5.1. G is a cycle This subsection deals with the case that G is a cycle. For clarity, denote G by C . The following algorithm, called Algorithm 2, will convert any minimum (F , B, R)-dominating set of C into a minimum (F , B, R)-dominating set D∗ of C that is suitable for performing mutual transferability. See Fig. 5 for an illustration. One claim is made. Claim 5.1. d(bi , bj ) > 2 (and hence d(ri , rj ) > 2) for any 1 ≤ i < j ≤ k except when {i, j} = {1, k}. Proof. Since D is a minimum (F , B, R)-dominating set of C , the set (S ∩ BI ) − T cannot be empty. Otherwise, we can find an (F , B, R)-dominating set of C having a cardinality smaller than |D|, a contradiction. Therefore b1 , b2 , . . . , bk and r1 , r2 , . . . , rk exist. By lines 7 and 8 in Algorithm 2, for each 1 ≤ t ≤ k, vertex rt is the successor of vertex bt (in clockwise ordering). Thus d(bi , bj ) ≤ 2 implies that bj ∈ N [ri ] and hence bj ̸ ∈ (S ∩ BI ) − T , which contradicts with bj ∈ (S ∩ BI ) − T (see line 6 in Algorithm 2). Since ri is the successor of bi and rj is the successor of bj (in clockwise ordering), we also have d(ri , rj ) > 2. □ Theorem 5.2. For any cycle C of n vertices and any minimum (F , B, R)-dominating set D of C , Algorithm 2 converts D into a ∗ minimum (F , B, R)-dominating set D∗ of C that satisfies Claim 5.1 in O(n) time. Moreover, D∗ = R ∪ {r1 , r2 , . . . , rk } and D → D∗ .
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Algorithm 2 Input: (1) A cycle C = (V , E) with its vertices v1 , v2 , . . . , vn listed in clockwise ordering and V being the disjoint union of F , B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. (2) A minimum (F , B, R)-dominating set D = R ∪ {vi1 , vi2 , . . . , vik } of C , where i1 < i2 < · · · < ik . Output: A minimum (F , B, R)-dominating set D∗ of C that satisfies Claim 5.1. ∗ 1: D ← R; 2: BI ← B − N(R); //BI = {important bound vertices, i.e., bound vertices not dominated by R} 3: for j = 1 to k do //replace vij with the vertex rj found below 4: S ← {vij −1 , vij , vij +1 }; //we assume vij −1 = vn for vij = v1 and vij +1 = v1 for vij = vn 5: if (j = 1) then T ← ∅; else T ← N [rj−1 ]; end if 6: vf ← the first-encountered bound vertex in (S ∩ BI ) − T under clockwise ordering vij −1 , vij , vij +1 ; 7: bj ← vf ; 8: rj ← vf +1 ; //b1 , b2 , . . . , bk and r1 , r2 , . . . , rk are used in proving correctness and mutually transferability 9: D∗ ← D∗ ∪ {rj }; 10: end for
Fig. 5. An illustration of Algorithm 2. R = {v5 }, the boxed node. (a) D = R ∪ {v2 , v8 , v10 }. (b) D∗ = R ∪ {v2 , v8 , v11 }.
Proof. Since D is a minimum (F , B, R)-dominating set of C , the set (S ∩ BI ) − T is non-empty. Therefore, during the execution of Algorithm 2, we can derive b1 , b2 , . . . , bk and r1 , r2 , . . . , rk . So D∗ = R ∪ {r1 , r2 , . . . , rk } is obvious. Since all the bound vertices dominated by D are dominated by D∗ and |D∗ | = |D|, it follows that D∗ is a minimum (F , B, R)-dominating set of C . Thus Algorithm 2 converts D into a minimum (F , B, R)-dominating set D∗ of C that satisfies Claim 5.1. We now analyze its time complexity. The most time-consuming steps in Algorithm 2 are lines 2, 5 and 6; all the other steps can be done in O(n) time. Since C is a cycle, each vertex has only two neighbors. Thus lines 2 and 5 takes O(n) time. Moreover, each vij has at most three bound vertices in its (S ∩ BI ) − T ; thus line 6 takes O(n) time. From the above, Algorithm 2 takes O(n) time. For mutual transferability, by moving vij to rj for j = 1, 2, . . . , k; each of the above moves is safe since every vertex in BI ∩ N [vi1 ] ∗
is dominated by r1 and every vertex in BI ∩ N [vij ], j ≥ 2, is dominated by rj−1 or rj . Thus D → D∗ .
□
Now we prove mutual transferability for (F , B, R)-dominating sets of a cycle. Theorem 5.3. Let C be a cycle and let its vertices be the disjoint union of F , B, and R. Let D1 be any (F , B, R)-dominating set of ∗ C , D∗ be the minimum (F , B, R)-dominating set of C produced by Algorithm 2, and ℓ = |D1 | − |D∗ |. Then D1 ∪ {1 · extra} → ∗ D ∪ {(ℓ + 1) · extra}. Proof. Let v1 , v2 , . . . , vn be the vertices of C listed in clockwise ordering around C . Note that by Algorithm 2 we obtain b1 , b2 , . . . , bk and r1 , r2 , . . . , rk with D∗ = R ∪ {r1 , . . . , rk }. Without loss of generality, we assume that an extra element in D1 ∪ {1 · extra} is located at v1 and assume that (ℓ + 1) extra elements in D∗ ∪ {(ℓ + 1) · extra} are also located at v1 . Let ∗ Y = D∗ ∪ {(ℓ + 1) · extra}. The process of performing D1 ∪ {1 · extra} → Y and its correctness are listed below. (See Fig. 6.) ◀ Standard Process for a Cycle ▶
1: For i = 1 to k − 1, let (ri , ri+1 ] denote the set of vertices in C between ri (exclusive) and ri+1 (inclusive) in a clockwise manner. Let (rk , r1 ) denote the set of vertices in C between rk and r1 (both exclusive) in a clockwise manner. (Take Fig. 6(b) as an example. Then, k = 3 and we have (r1 , r2 ] = (v2 , v8 ] = {v3 , v4 , v5 , v6 , v7 , v8 }, (r2 , r3 ] = (v8 , v11 ] = {v9 , v10 , v11 }, and (r3 , r1 ) = (v11 , v2 ) = {v1 }.) (1)
2: Move an extra element in D1 ∪ {1 · extra} to r1 . Such a move is safe since the moved element is extra. Let D1 be the resultant (F , B, R)-dominating set.
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∗
Fig. 6. The process of D1 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra}, where D1 = {v1 , v4 , v5 , v8 , v10 }, D∗ = {v2 , v5 , v8 , v11 }, ℓ = 1, and R = {v5 } (1) (2) (3) (the boxed node). (a) D1 ∪ {1 · extra}. (b) D1 = {v1 , v2 , v4 , v5 , v8 , v10 }. (c) D1 = {v1 , v2 , v5 , v8 , v8 , v10 }. (d) D1 = {v1 , v2 , v5 , v8 , v11 , v11 }. (e) (4)
D1 = {v2 , v5 , v8 , v11 , v1 , v1 } = D∗ ∪ {2 · extra}. (i)
3: For i = 1 to k − 1 do the following. For all of the elements in D1 that are located at ri , keep one of them at ri and move all of (i) the others to ri+1 . Then, move all of the elements in D1 that are located in (ri , ri+1 ] to ri+1 . By Claim 5.1, d(bi , bi+1 ) > 2; (i) thus N [bi+1 ] ⊆ (ri , ri+1 ], meaning that at least one element in D1 is located in (ri , ri+1 ] and therefore after the moves, at least one element is located at ri+1 . Notice that each of the above moves is safe since all the bound vertices in (ri , ri+1 ] (i+1) are dominated by {ri , ri+1 } ∪ R. Let the resultant (F , B, R)-dominating set be D1 . (k)
4: For all of the elements in D1 that are located at rk , keep one of them at rk and move all of the others to v1 . Then, move all (k) of the elements in D1 that are located in (rk , r1 ) to v1 . Each of the above moves is safe since all the bound vertices in (k+1) (k+1) is exactly Y . The set D1 (rk , r1 ) are dominated by {rk , v1 , r1 } ∪ R. Let the resultant (F , B, R)-dominating set be D1 and one extra element in Y is located at v1 . □ 5.2. G has a cycle end-block Let C be a cycle end-block of G and x be its unique cut-vertex. Following reference [16], we use CF and CR to denote C with x being relabeled as free and required, respectively, and DCF , DC and DCR to denote the minimum (F , B, R)-dominating set obtained by MCYCLE(CF ), MCYCLE(C ) and MCYCLE(CR ), respectively. Reference [16] says that |DCF | ≤ |DC | ≤ |DCR | < |DCF | + 1. There is a typo and the statement should be |DCF | ≤ |DC | ≤ |DCR | ≤ |DCF | + 1. Let D1 be any (F , B, R)-dominating set of G. Let C be any cycle end-block of G and x its unique cut-vertex. We use C − x to denote the graph obtained by deleting x from C . Set AF = D1 ∩ V (C − x)
and A = D1 ∩ V (C )
for easy writing. One lemma is needed. Lemma 5.4. Let G be a cactus graph with a cycle end-block C , x be the unique cut-vertex in C , and D1 be any (F , B, R)-dominating set of G. Then:
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• If |DCF | < |DC |, then |AF | ≥ |DCF |. • If |DCF | = |DC | < |DCR |, then |A | ≥ |DC |. • If |DCF | = |DC | = |DCR |, then |A | ≥ |DCR |. Proof. If |DCF | < |DC |, then AF ∪ {x} is an (F , B, R)-dominating set of C and DC is a minimum (F , B, R)-dominating set of C ; thus we have |AF | + 1 ≥ |DC | > |DCF |, that is, |AF | ≥ |DCF |. If |DCF | = |DC |, then A is an (F , B, R)-dominating set of CF and DCF is a minimum (F , B, R)-dominating set of CF ; thus we have |A | ≥ |DCF | = |DC |. Moreover, if |DCF | = |DC | = |DCR |, then we have |A | ≥ |DCF | = |DCR |. □ By Lemma 5.4, when |DCF | < |DC |, we have |AF | ≥ |DCF |; thus we define:
• D1 (DCF ) = (D1 − AF ) ∪ DCF ∪ {x, x, . . . , x}
(total number of x is |AF |−|DCF |).
As an example, if DCF = {v3 , v6 , v9 } and |AF | = 5, then DCF ∪ {x, x, . . . , x} is {v3 , v6 , v9 , x, x}. By Lemma 5.4, when |DCF | = |DC |, we have |A | ≥ |DC | or |A | ≥ |DCR |; thus we define:
• D1 (DC ) = (D1 − A ) ∪ DC ∪ {x, x, . . . , x} (total number of x is |A | − |DC |) x is |A | − |DC |) • D1 (DCR ) = (D1 − A ) ∪ DCR ∪ {x, x, . . . , x} (total number of x is |A | − |DCR |). The followings are clear:
• DCF ⊆ D1 (DCF ), DC ⊆ D1 (DC ), DCR ⊆ D1 (DCR ) • |D1 (DCF )| = |D1 (DC )| = |D1 (DCR )| = |D1 | • Each of D1 (DCF ), D1 (DC ), and D1 (DCR ) is an (F , B, R)-dominating set of G. Lemma 5.5. Let G be a cactus graph with a cycle end-block C , x be the unique cut-vertex in C , and D1 be any (F , B, R)-dominating set of G. ∗
• If |DCF | < |DC |, then D1 ∪ {1 · extra} → D1 (DCF ) ∪ {1 · extra}. ∗ • If |DCF | = |DC | < |DCR |, then D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra}. ∗ • If |DCF | = |DC | = |DCR |, then D1 ∪ {1 · extra} → D1 (DCR ) ∪ {1 · extra}. Proof. Let v1 (= x), v2 , . . . , vn be the vertices of C listed in clockwise ordering around C . Without loss of generality, we assume that an extra element in D1 ∪ {1 · extra} is located at v1 .
• If |DCF | < |DC |, let Z = CF , X = DCF and Y = D1 (DCF ) ∪ {1 · extra}. • If |DCF | = |DC | < |DCR |, let Z = C , X = DC and Y = D1 (DC ) ∪ {1 · extra}. • If |DCF | = |DC | = |DCR |, let Z = CR , X = DCR and Y = D1 (DCR ) ∪ {1 · extra}. ∗
Let the first input of Algorithm 2 be Z and the second input be X , and run Algorithm 2. Then, the process of D1 ∪{1 · extra} → Y and its correctness follows from Standard Process for a Cycle in Theorem 5.3. □ 5.3. The general case Now we deal with the general case of cactus graphs. ∗
Theorem 5.6. For any cactus graph G, if D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |, then D1 ∪{1 · extra} → D2 ∪ {1 · extra}. Proof. If G is a tree, then this theorem follows from Theorem 4.2 (actually, no extra element is needed). In what follows, we assume that G is not a tree and is of order at least three. We prove by induction on the number m of blocks of G. If m = 1, ∗ ∗ then G is a cycle. By Theorem 5.3, D1 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra} and D2 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra} ∗ both are true, where D be the minimum (F , B, R)-dominating set of G produced by Algorithm 2 and ℓ = |D1 | − |D∗ |. ∗ ∗ Notice that D2 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra} implies D∗ ∪ {(ℓ + 1) · extra} → D2 ∪ {1 · extra}. Combining ∗ ∗ ∗ ∗ ∗ D1 ∪ {1 · extra} → D ∪ {(ℓ + 1) · extra} and D ∪ {(ℓ + 1) · extra} → D2 ∪ {1 · extra} obtains D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, meaning that the theorem holds when m = 1. In the following, let m ≥ 2 and assume that the theorem holds for cactus graphs having m − 1 blocks. There are two cases. Case 1: G has an end-block that is an edge. Then, let u be the vertex of degree 1 in this end-block and let x be the unique neighbor of u. Then, x must be a cut-vertex in G. Depending on the label of u, there are three subcases. 1.1: u is required. Then, at least one element in D1 must be u. Similarly, at least one element in D2 must be u. Let G′ = G − u and relabel x as free if its label in G is bound. Let D′′1 (respectively, D′′2 ) be the multiset obtained from D1 (respectively, D2 ) by keeping only one element at u and moving all of the other elements that are
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located at u to x. Let D′1 and D′2 be the multiset obtained from D′′1 and D′′2 by deleting the unique u in D′′1 and D′′2 , respectively. 1.2: u is free. Then, let G′ = G − u. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (respectively, D2 ) by moving every element located at u to x. 1.3: u is bound. Then, since u is bound, at least one element in D1 must be u or x. Similarly, at least one element in D2 must be u or x. Let G′ = G − u and relabel x as required. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (respectively, D2 ) by moving every element located at u to x. In any of the subcases 1.1, 1.2, and 1.3, D′1 and D′2 are (F , B, R)-dominating sets of G′ with |D′1 | = |D′2 |. Since u and x are adjacent, all the moves used in obtaining D′1 and D′2 are safe. Moreover, G′ is a cactus graph with m − 1 blocks. ∗
∗
By the induction hypothesis, D′1 ∪ {1 · extra} → D′2 ∪ {1 · extra}. Thus D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. Case 2: Every end-block in G is a cycle. Then, pick up an arbitrary end-block C in G and let x be the unique cut-vertex in C . Depending on the label of x, there are three subcases. 2.1: x is required. First we execute MCYCLE(C ) to obtain DC ; then use DC to obtain D1 (DC ) and D2 (DC ). Next we run Algorithm 2 with C and DC as its inputs, and we follow Standard Process for a Cycle in Theorem 5.3 to obtain ∗ ∗ D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra} and D2 ∪ {1 · extra} → D2 (DC ) ∪ {1 · extra}. Let G′ = G − (C − x). Let D′1 ′ (respectively, D2 ) be the multiset obtained from D1 (DC ) (respectively, D2 (DC )) by deleting all of its elements appearing in C − x. 2.2: x is free. In this case, |DCF | = |DC | occurs. Therefore, we first execute MCYCLE(C ) and MCYCLE(CR ) to obtain DC and DCR ; then use DC and DCR to obtain D1 (DC ), D2 (DC ), D1 (DCR ), and D2 (DCR ). There are two subcases. ∗
2.2.1: |DCF | = |DC | < |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra} and ∗
D2 ∪ {1 · extra} → D2 (DC ) ∪ {1 · extra}. Let G′ = G − (C − x). Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DC ) (respectively, D2 (DC )) by deleting all of its elements appearing in C − x. ∗ 2.2.2: |DCF | = |DC | = |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DCR ) ∪ {1 · extra} and ∗
D2 ∪ {1 · extra} → D2 (DCR ) ∪ {1 · extra}. Let G′ = G − (C − x) and relabel x in G′ as required. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DCR ) (respectively, D2 (DCR )) by deleting all of its elements appearing in C − x. 2.3: x is bound. Then, we first execute MCYCLE(CF ), MCYCLE(C ) and MCYCLE(CR ) to obtain DCF , DC and DCR ; then use DCF , DC and DCR to obtain D1 (DCF ), D2 (DCF ), D1 (DC ), D2 (DC ), D1 (DCR ), and D2 (DCR ). ∗
∗
2.3.1: |DCF | < |DC |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DCF ) ∪ {1 · extra} and D2 ∪ {1 · extra} → D2 (DCF ) ∪ {1 · extra}. Let G′ = G − (C − x). Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DCF ) (respectively, D2 (DCF )) by deleting all of its elements appearing in C − x. ∗
2.3.2: |DCF | = |DC | < |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra} and ∗
D2 ∪{1 · extra} → D2 (DC ) ∪{1 · extra}. Let G′ = G − (C − x) and relabel x in G′ as free. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DC ) (respectively, D2 (DC )) by deleting all of its elements appearing in C − x. ∗ 2.3.3: |DCF | = |DC | = |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DCR ) ∪ {1 · extra} and ∗
D2 ∪ {1 · extra} → D2 (DCR ) ∪ {1 · extra}. Let G′ = G − (C − x) and relabel x in G′ as required. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DCR ) (respectively, D2 (DCR )) by deleting all of its elements appearing in C − x. In any of the subcases 2.1, 2.2.1, 2.2.2, 2.3.1, 2.3.2, and 2.3.3, D′1 and D′2 are (F , B, R)-dominating sets of G′ with |D′1 | = ∗
|D′2 |. Moreover, G′ is a cactus graph with m − 1 blocks. By the induction hypothesis, D′1 ∪ {1 · extra} → D′2 ∪ {1 · extra}. ∗ Thus D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. We now complete the proof. □ By letting B = V (G) and F = R = ∅, we have the following corollary. ∗
Corollary 5.7. If G is a cactus graph in which D1 and D2 are two dominating sets with |D1 | = |D2 |, then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. 6. Concluding remarks In this paper, we have proved any strongly chordal graph has the strong duality equality for (F , B, R)-domination. We ∗ ∗ have also shown that if G is connected strongly chordal then D1 → D2 , and if G is a cactus graph then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, where D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |. A preliminary version of this paper was in [7].
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For any connected graph G, we may try to determine the least number ℓ(G) of extra elements required to guarantee ∗ mutually transferability. That is, if D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |, then D1 ∪ {ℓ · extra} → D2 ∪ {ℓ · extra} for ℓ ≥ ℓ(G). For example, ℓ(G) = 0 when G is a connected strongly chordal graph, and ℓ(G) = 1 when G is a cactus graph. We conjecture that any connected graph G has ℓ(G) ≤ 1. In fact, for cycles Cn , we yield the following partial results: if all vertices are labeled as bound (or equivalently, ordinary domination), then ℓ(Cn ) = 1 if n = 3r + 6 and ℓ(Cn ) = 0 otherwise; if there is a vertex labeled as required, then ℓ(Cn ) = 0; and if no vertex is labeled as required and there are adjacent vertices labeled as free, then ℓ(Cn ) = 0. Acknowledgments The authors would like to thank Prof. Gerard Jennhwa Chang and Prof. Sheng-Chyang Liaw for their valuable comments that greatly improved the presentation of this paper. The authors would also like to express their deepest gratitude to the anonymous reviewers for their constructive comments and time spent to analyze this paper. References [1] R.P. Anstee, M. Farber, Characterizations of totally balanced matrices, J. Algorithms 5 (2) (1984) 215–230. [2] K.S. Booth, J.H. Johnson, Dominating sets in chordal graphs, SIAM J. Comput. 11 (1) (1982) 191–199. [3] G.J. Chang, k-domination and graph covering problems (Ph.D thesis), School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1982. [4] G.J. Chang, Algorithmic aspects of domination in graphs, in: D.-Z. Du, P.M. Pardalos (Eds.), Handbook of Combinatorial Optimization, vol. 3, 1998, pp. 339–405. [5] G.J. Chang, G.L. Nemhauser, The k-domination and k-stability problems in sun-free chordal graphs, SIAM J. Algebr. Discrete Methods 5 (3) (1984) 332–345. [6] L. Chen, C. Lu, Z. Zeng, A linear-time algorithm for paired-domination problem in strongly chordal graphs, Inform. Process. Lett. 110 (1) (2009) 20–23. [7] K.T. Chu, Mutual transferability of dominating sets in strongly chordal graphs and cactus graphs, (Master thesis), National Tsing Hua University, Hsinchu, Taiwan, 2018. [8] E.J. Cockayne, S. Goodman, S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1) (1975) 41–44. [9] D.G. Corneil, L.K. Stewart, Dominating sets in perfect graphs, Discrete Math. 86 (1–3) (1990) 145–164. [10] K. Das, Some algorithms on cactus graphs, Ann. Pure Appl. Math. 2 (2) (2012) 114–128. [11] M. Farber, Applications of linear programming duality to problems involving independence and domination (Ph.D Thesis), Rutgers University, New Brunswick, NJ, 1982, also issued as Technical Report 81-13, Computing Science Department, Simon Fraser University, 1981. [12] M. Farber, Characterizations of strongly chordal graphs, Discrete Math. 43 (2–3) (1983) 173–189. [13] M. Farber, Domination, independent domination and duality in strongly chordal graphs, Discrete Appl. Math. 7 (2) (1984) 115–130. [14] S. Fujita, A tight bound on the number of mobile servers to guarantee transferability among dominating configurations, Discrete Appl. Math. 158 (8) (2010) 913–920. [15] P.A. Golovach, P. Heggernes, N. Lindzey, R.M. McConnell, V.F. dos Santos, J.P. Spinrad, J.L. Szwarcfiter, On recognition of threshold tolenrance graphs and their complements, Discrete Appl. Math. 216 (1) (2017) 171–180. [16] S.T. Hedetniemi, R. Laskar, J. Pfaff, A linear algorithm for finding a minimum dominating set in a cactus, Discrete Appl. Math. 13 (2–3) (1986) 287–292. [17] A. Kolen, Solving covering problems and the uncapacitated plant location problem on trees, Eur. J. Oper. Res. 12 (3) (1983) 266–278. [18] R. Laskar, J. Pfaff, S.M. Hedetniemi, S.T. Hedetniemi, On the algorithmic complexity of total domination, SIAM J. Algebr. Discrete Methods 5 (3) (1984) 420–425. [19] A. Lubiw, Doubly lexical orderings of matrices, SIAM J. Comput. 16 (5) (1987) 854–879. [20] R. Nevries, C. Rosenke, Characterizing and computing the structure of clique intersections in strongly chordal graphs, Discrete Appl. Math. 181 (2015) 221–234. [21] R. Paige, R.E. Tarjan, Three partition refinement algorithms, SIAM J. Comput. 16 (6) (1987) 973–989. [22] J. Sawada, J.P. Spinrad, From a simple elimination ordering to a strong elimination ordering in linear time, Inform. Process. Lett. 86 (6) (2003) 299–302.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Mutual transferability for (F , B, R)-domination on strongly chordal graphs and cactus graphs✩ ∗
Kuan-Ting Chu a , Wu-Hsiung Lin b , , Chiuyuan Chen b a b
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
article
info
Article history: Received 30 April 2018 Received in revised form 22 October 2018 Accepted 28 December 2018 Available online 23 January 2019 Keywords: Domination Stability Transferability Strongly chordal graphs Cactus graphs
a b s t r a c t This paper studies a variation of domination in graphs called (F , B, R)-domination. Let G = (V , E) be a graph and V be the disjoint union of F , B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. An (F , B, R)dominating set of G is a subset D ⊆ V such that R ⊆ D and each vertex in B − D is adjacent to some vertex in D. An (F , B, R)-2-stable set of G is a subset S ⊆ B such that S ∩ N(R) = ∅ and every two distinct vertices x and y in S have distance d(x, y) > 2. We prove that if G is strongly chordal, then αF ,B,R,2 (G) = γF ,B,R (G)−|R|, where γF ,B,R (G) is the minimum cardinality of an (F , B, R)-dominating set of G and αF ,B,R,2 (G) is the maximum cardinality of an (F , B, R)∗
2-stable set of G. Let D1 → D2 denote D1 being transferable to D2 . We prove that if G is a connected strongly chordal graph in which D1 and D2 are two (F , B, R)-dominating sets ∗ with |D1 | = |D2 |, then D1 → D2 . We also prove that if G is a cactus graph in which D1 and D2 ∗ are two (F , B, R)-dominating sets with |D1 | = |D2 |, then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, where ∪{1 · extra} means adding one extra element. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Domination and its variations in graphs have been widely studied in the literature and they have numerous applications in computer networks. Especially, they can be used in server allocation to ensure the Quality of Service (QoS) of a network. Let G = (V , E) be a graph. The neighborhood N(v ) of a vertex v is the set to v ; the closed neighborhood ⋃of all vertices adjacent⋃ N [v] of v is N [v] = N(v ) ∪ {v}. For any subset S ⊆ V , define N(S) = v∈S N(v ) and N [S ] = v∈S N [v]. A vertex v is said to dominate all vertices in N [v]. A subset D ⊆ V is a dominating set of G if every vertex ⋃ in V − D is adjacent to at least one vertex in D. Chang [4] pointed out that this is equivalent to N [v] ∩ D ̸ = ∅ for all v ∈ V or u∈D N [u] = V . A subset S ⊆ V is a 2-stable set of G if every two distinct vertices x and y in S have d(x, y) > 2. The domination number γ (G) of G is the minimum cardinality of a dominating set of G. The 2-stability number α2 (G) of G is the maximum cardinality of a 2-stable set of G. It is easy to see that any graph G has (Weak Duality Inequality for Domination): α2 (G) ≤ γ (G). Chang [4] used an algorithmic approach to show that any tree T has (Strong Duality Equality for Domination): α2 (T ) = γ (T ). Such a primal–dual approach was used by Farber [13] and Kolen [17] for weighted domination on strongly chordal graphs. Some definitions and notations that follow the paper of Fujita [14] must be introduced. Let D1 and D2 be two dominating sets of G. We say that D1 is single-step transferable to D2 , denoted as D1 → D2 , if there exist u ∈ D1 and v ∈ D2 such that ✩ This research was partially supported by the Ministry of Science and Technology of Taiwan under grants MOST-106-2115-M-009-005 and MOST-1062115-M-009-007. ∗ Corresponding author. E-mail address: [email protected] (W.-H. Lin). https://doi.org/10.1016/j.dam.2018.12.034 0166-218X/© 2019 Elsevier B.V. All rights reserved.
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∗
Fig. 1. Let D1 = {a, b, e}, D2 = {a, c , f }, and D3 = {a, c , e}. Then, D1 → D3 , D3 → D2 , and D1 → D2 .
∗
uv ∈ E and D1 − {u} = D2 − {v}. We write D1 → D2 if D1 can be transferred to D2 through a sequence of single-step transfers ∗ (also called a sequence of safe moves in [14]). By definition, D1 → D2 implies that D2 → D1 and |D1 | = |D2 |; also, D1 → D2 ∗ implies that D2 → D1 and |D1 | = |D2 |. Fig. 1 gives an illustration. For technical reasons, [14] assumes that a dominating set of G is a dominating multiset (which is like a set except that its members need not be distinct); in this paper, we make the same assumption. Reference [14] studies the following server allocation problem: Given two dominating sets D1 and D2 of a graph G, can we transfer D1 to D2 through a sequence of safe moves? The main results of [14] are as follows. ∗
• If T is a tree with n vertices in which D1 and D2 are two dominating sets with |D1 | = |D2 | ≥ ⌊ 2n ⌋, then D1 → D2 . • If G is a Hamiltonian graph with n vertices in which D1 and D2 are two dominating sets with |D1 | = |D2 | ≥ ⌈ n+3 1 ⌉, then ∗ D1 → D2 . The purpose of this paper is to study (F , B, R)-domination (also called mixed domination) for strongly chordal graphs and cactus graphs. Let G = (V , E) be a graph and suppose V is the disjoint union of three sets F , B, and R, where F means ‘‘Free’’, consisting of free vertices, B means ‘‘Bound’’, consisting of bound vertices, and R means ‘‘Required’’, consisting of required vertices. An (F , B, R)-dominating set of G is a subset D ⊆ V such that R ⊆ D and each vertex in B − D is adjacent to some vertex in D; free vertices need not be dominated by D but may be included in D to dominate vertices. (See [8,16].) Notice that ordinary domination is the same as (F , B, R)-domination when B = V (G), F = R = ∅. An (F , B, R)-2-stable set of G is a subset S ⊆ B such that S ∩ N(R) = ∅ and every two distinct vertices x and y in S have d(x, y) > 2. The (F , B, R)-domination number γF ,B,R (G) of G is the minimum cardinality of an (F , B, R)-dominating set of G. The (F , B, R)-2-stability number αF ,B,R,2 (G) of G is the maximum cardinality of an (F , B, R)-2-stable set of G. By replacing ‘‘dominating set’’ with ‘‘(F , B, R)-dominating set’’ ∗ in the definitions of D1 → D2 and D1 → D2 , mutual transferability among (F , B, R)-dominating sets D1 and D2 of G can be defined and we omit the details. We have one remark for (F , B, R)-dominating sets. Up to now, finding a minimum (F , B, R)-dominating set is regarded as an approach (a labeling method) for deriving a minimum dominating set. The final goal is a minimum dominating set, not a minimum (F , B, R)-dominating set. However, as can be seen from Fig. 2 and the following scenario, a minimum (F , B, R)dominating set is more flexible than a minimum dominating set when we want to model a real-world problem.
• Some of the cities in Taiwan are the most important; for example, Taipei, Hsinchu, and Kaohsiung. We need to allocate a server for each of them and therefore mark each of them as required.
• Some of the cities in Taiwan are important but not the most important; for example, Taoyuan, Taichung, Tainan, and so on. We do not need to allocate a server for each of them but we need to ensure that each of them has a server within its closed neighborhood. We thus mark each of them as bound. • Some of the cities in Taiwan are not so important and we do not even have to make each of them have a server within its closed neighborhood. We therefore mark each of them as free. It is easy to see the following inequality. Weak Duality Inequality for (F , B, R) -Domination: αF ,B,R,2 (G) ≤ γF ,B,R (G) − |R| for any graph G. The inequality can be strict, as shown by the 5-cycle C5 (with B being the set of all vertices and F = R = ∅) that αF ,B,R,2 (C5 ) = 1 < 2 = γF ,B,R (C5 ) − |R|. Main results of this paper are:
• Any strongly chordal graph G has (Strong Duality Equality for (F , B, R)-Domination): αF ,B,R,2 (G) = γF ,B,R (G) − |R|. • If G is a connected strongly chordal graph in which D1 and D2 are two (F , B, R)-dominating sets with |D1 | = |D2 |, then ∗ D1 → D2 . ∗ • If G is a cactus graph in which D1 and D2 are two (F , B, R)-dominating sets with |D1 | = |D2 |, then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. Definitions of strongly chordal graphs, cactus graphs, and ∪{1 · extra} will be given later. It is well known that strongly chordal graphs is a subclass of chordal graphs. On the other hand, trees, block graphs, interval graphs, and directed path
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Fig. 2. Map of Taiwan.
graphs are subclasses of strongly chordal graphs. See [2,9] for definitions of these graphs. Containment relations between these graphs can be shown below: tree interval graph
⊂ ⊂
block graph directed path graph
⊂ ⊂
strongly chordal graph strongly chordal graph
⊂ ⊂
chordal graph chordal graph
It follows that our second main result applies to trees, block graphs, interval graphs, and directed path graphs. Thus it improves the first main result of [14], which only applies to trees and dominating sets of cardinality ≥ ⌊ 2n ⌋. Every cycle C3r +6 , r ≥ 0, is a cactus graph. However, no two distinct minimum dominating sets of C3r +6 are mutually transferable; therefore at least one extra element has to be added. Take the C6 shown in Fig. 1 as an example. Both {b, e} and {c , f } are minimum dominating sets of C6 but they are not mutually transferable. After the extra element a is added to {b, e} and also ∗ to {c , f }, we have {b, e} ∪ {a} → {c , f } ∪ {a}. Our third main result (which adds only one extra element) is therefore the best possible. This paper is organized as follows. Section 2 gives preliminaries. Section 3 establishes the strong duality equality for (F , B, R)-domination on strongly chordal graphs. Sections 4 and 5 prove mutual transferability for (F , B, R)-domination on connected strongly chordal graphs and on cactus graphs, respectively. Concluding remarks are drawn in Section 6. 2. Preliminaries This paper considers undirected graphs G = (V , E) without loops or multiple edges. We now define strongly chordal graphs. Suppose G = (V , E) is a graph and v1 , v2 , . . . , vn is an ordering of V (G). Set Ni [vj ] = N [vj ] ∩ {vi , vi+1 , . . . , vn } for easy writing. Two definitions are given. Definition 1. A strong elimination ordering of a graph G = (V , E) is an ordering v1 , v2 , . . . , vn of V such that for each i, j, and k, if i ≤ j ≤ k and vj , vk ∈ N [vi ], then Ni [vj ] ⊆ Ni [vk ]. Definition 2. A graph is strongly chordal if it admits a strong elimination ordering. For convenience, throughout this paper, vi∗ is used to denote the largest-indexed vertex in N [vi ]. The class of strongly chordal graphs was introduced by Farber in [11]; see also [12,13]. Chang and Nemhauser [3,5] independently discovered this class of graphs. Reference [12] provides a polynomial-time algorithm to recognize strongly chordal graphs and to construct strong elimination orderings; see also [1,19–21]. Actually, several versions of strong elimination orderings have been proposed (see [6,12,13,15,22]) and can be proven to be equivalent. In this paper, we assume that a strong elimination ordering is always available when the given graph is strongly chordal. We now are ready to define the notion ∪{1 · extra}. To do this, we actually define the notion D ∪ {ℓ · extra}. Intuitively, when an (F , B, R)-dominating set of G contains more than γF ,B,R (G) elements, it may contain elements whose deletion will not affect the (F , B, R)-domination and we call these elements extra elements. More precisely, suppose D is already an (F , B, R)dominating set of G and {s1 , s2 , . . . , sℓ } is a subset of V (G). Then, D ∪ {s1 , s2 , . . . , sℓ } is an (F , B, R)-dominating set of G and s1 , s2 , . . . , sℓ are extra elements. When the names of s1 , s2 , . . . , sℓ are unimportant, we simply write D ∪ {s1 , s2 , . . . , sℓ } as D ∪ {ℓ · extra}. By setting ℓ = 1 into D ∪ {ℓ · extra}, we have D ∪ {1 · extra} and the meaning of the notion ∪{1 · extra} is now ∗ clear. Notice that when D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, a sequence of single-step transfers from D1 to D2 can contain an (F , B, R)-dominating set of G that has no extra element; D3 in Fig. 1 is an example. Before closing this section, we briefly describe the background of the server allocation problem studied by Fujita [14]. The purpose of this problem is to guarantee a continuous service to all users of a network by a collection of mobile servers distributed over the network. Let D be the (multi)set of nodes that have mobile servers. Then, all users can receive the service
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of the network if D is a dominating (multi)set of the network’s corresponding graph G. Since mobile servers can move, to guarantee a continuous service to all users, the change of servers’ locations must obey three rules:
• A mobile server can move from a node to any adjacent node in a single step. • In each step, at most one mobile server can move. • Each move does not leave any node not dominated after the move. Notice that during the move of a mobile server, some nodes might be not dominated. Also notice that for (F , B, R)-domination, only bound vertices need to be dominated. A move of a mobile server is said to be safe if it obeys the above rules; this definition is crucial for the rest of this paper. Based on the above rules, [14] studies the mutual transferability among ∗ dominating sets D1 and D2 of G and asks if D1 → D2 . 3. Strong duality equality for (F , B, R)-domination on strongly chordal graphs The study of domination in strongly chordal graphs by Farber [11,13], Chang and Nemhauser [5] and Kolen [17] also leads to the strong duality equality γ (G) = α2 (G). On the other hand, the approaches of labeling algorithms for (F , B, R)domination (for example, Cockayne et al. [8] for trees, Laskar et al. [18] for total domination, Hedetniemi et al. [16] for cactus graphs, Chang [4] for strongly chordal graphs) are somehow ad hoc. In this section we give a primal–dual approach for the (F , B, R)-domination on strongly chordal graphs, with the strong duality equality as a by product. Using the same spirit, we give a mutual transferability in the next section. In particular, we show that αF ,B,R,2 (G) = γF ,B,R (G) − |R| for any strongly chordal graph G. We do this by proposing an algorithm, called Algorithm 1, to find both a minimum (F , B, R)-dominating set D∗ and a maximum (F , B, R)-2-stable set S ∗ of G. Here we briefly describe the idea behind Algorithm 1. Suppose v1 , v2 , . . . , vn is a strong elimination ordering of G and vi is a bound vertex. Then, at least one vertex in N [vi ] has to be put into D∗ to dominate vi . Recall that we denote by vi∗ the largest-indexed vertex in N [vi ]. Therefore suppose N [vi ] = {vi1 , vi2 , . . . , viℓ }, where i1 < i2 < · · · < iℓ . Then, by definition, vi∗ = viℓ . Since Ni [vi1 ] ⊆ Ni [vi2 ] ⊆ · · · ⊆ Ni [vi∗ ], Algorithm 1 puts vi∗ into D∗ . Besides, Algorithm 1 puts vi into S ∗ . Notice that Farber [13] has proposed an algorithm to find a minimum weighted dominating set for any strongly chordal graph. Farber’s algorithm can find a minimum dominating set, but not a minimum (F , B, R)-dominating set. Algorithm 1 Input: A strongly chordal graph G = (V , E) with a strong elimination ordering v1 , v2 , . . . , vn and a partition F , B, R of V , where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. Output: A minimum (F , B, R)-dominating set D∗ and a maximum (F , B, R)-2-stable set S ∗ of G. ∗ 1: D ← R; ∗ 2: S ← ∅; 3: relabel all the bound vertices in N(R) as free; 4: k ← 0; 5: for i = 1 to n do 6: if (vi is bound) then 7: k ← k + 1; 8: bk ← v i ; 9: rk ← vi∗ ; //b1 , b2 , . . . , bk and r1 , r2 , . . . , rk are used in proving correctness and mutually transferability 10: relabel vi∗ as required; 11: relabel all the bound vertices in N(vi∗ ) as free; 12: D∗ ← D∗ ∪ {vi∗ }; 13: S ∗ ← S ∗ ∪ {vi }; 14: end if 15: end for Fig. 3 shows an example of Algorithm 1. Two claims are made. Claim 3.1. N [R] ∩ {b1 , b2 , . . . , bk } = ∅ and R ∩ {r1 , r2 , . . . , rk } = ∅. Proof. During the execution of line 3 of Algorithm 1, all the bound vertices in N(R) are relabeled as free. Since b1 , b2 , . . . , bk are identified during the execution of line 8 of Algorithm 1, none of b1 , b2 , . . . , bk is a bound vertex in N(R). Hence N [R] ∩ {b1 , b2 , . . . , bk } = ∅. Thus R cannot dominate b1 , b2 , . . . , bk . Therefore R ∩ {r1 , r2 , . . . , rk } = ∅. □ Claim 3.2. d(bi , bj ) > 2 for any 1 ≤ i < j ≤ k. Proof. Notice that ri is the largest-indexed vertex in N [bi ]. We first prove that d(bi , bj ) = 1 is impossible. Suppose d(bi , bj ) = 1. Then, bj ∈ N [bi ]. By the property of a strongly chordal graph, bj ∈ N [ri ] must occur. Thus once
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Fig. 3. (a) A strongly chordal graph with F = ∅, B = {v1 , v2 , . . . , v9 }, R = ∅, and strong elimination ordering v1 , v2 , . . . , v9 . (b) Identifying b1 and r1 . (c) Identifying b2 and r2 . (d) Identifying b3 and r3 . Output will be D∗ = {r1 , r2 , r3 } = {v3 , v6 , v9 } and S ∗ = {b1 , b2 , b3 } = {v1 , v5 , v8 }.
Algorithm 1 identifies ri (in order to dominate bi ), it relabels the vertex that corresponds to bj as free (see line 11 of Algorithm 1). Hence the vertex that corresponds to bj is no longer a bound vertex; so d(bi , bj ) = 1 is impossible. We now prove that d(bi , bj ) = 2 is also impossible. Suppose d(bi , bj ) = 2. Then, bi and bj have a common neighbor vt0 . Let vt1 and vt2 be the vertices corresponding to bi and bj , respectively. It is clear that t1 < t2 . There are two cases. Case 1: t0 < t1 . Then, since vt1 , vt2 ∈ N [vt0 ], by the property of the strong elimination ordering, Nt0 [vt1 ] ⊆ Nt0 [vt2 ] must occur. Since bi ∈ Nt0 [vt1 ] and Nt0 [vt2 ] ⊆ N [bj ], it follows that bi ∈ N [bj ], contradicting with the assumption that d(bi , bj ) = 2. Case 2: t1 < t0 . Then, either vt0 and ri are the same vertices or they are different. In the former case, vt2 ∈ N [ri ] is assured. In the latter case, vt0 must proceed ri in the strong elimination ordering. Then, since vt0 , ri ∈ N [vt1 ], by the property of the strong elimination ordering, we have Nt1 [vt0 ] ⊆ Nt1 [ri ]. Since vt2 ∈ Nt1 [vt0 ] and Nt1 [ri ] ⊆ N [ri ], it follows that vt2 ∈ N [ri ] is also assured. Hence, once Algorithm 1 identifies ri (in order to dominate bi ), it relabels vt2 as free, contradicting with the assumption that bj is bound. We thus have this claim. □ The correctness of Algorithm 1 and the strong duality equality for (F , B, R)-domination can now be proved. Theorem 3.3. By using the adjacency-list representation of graphs, Algorithm 1 gives a minimum (F , B, R)-dominating set D∗ and a maximum (F , B, R)-2-stable set S ∗ of a strongly chordal graph G = (V , E) in O(|V | + |E |) time provided that a strong elimination ordering is given. Moreover, the strong duality equality αF ,B,R,2 (G) = γF ,B,R (G) − |R| holds. Proof. It is clear that D∗ = R ∪ {r1 , r2 , . . . , rk } and S ∗ = {b1 , b2 , . . . , bk }. Since each bi is a bound vertex, S ∗ ⊆ B follows. By Claim 3.1, S ∗ ∩ N [R] = ∅. By Claim 3.2, every two distinct bi and bj in S ∗ have distance d(bi , bj ) > 2. Therefore S ∗ is an (F , B, R)-2-stable set of G. That D∗ is an (F , B, R)-dominating set of G is due to (i)∼(iii) below. (i) Algorithm 1 relabels a bound vertex vi as free only when vi is adjacent to some required vertex (see lines 3 and 11). (ii) Algorithm 1 relabels a bound vertex vi as required only when vi itself is the largest-indexed vertex in N [vi ] or it is the largest-indexed neighbor of a bound vertex proceeding it (see lines 6, 9 and 10). (iii) Algorithm 1 never relabels a required vertex and it includes every required vertex into D∗ (see lines 1, 10, and 12). Since S ∗ is an (F , B, R)-2-stable set and D∗ is an (F , B, R)-dominating set of G, we have |S ∗ | ≤ αF ,B,R,2 (G) and γF ,B,R (G) ≤ |D∗ |. By the weak duality inequality for (F , B, R)-domination, αF ,B,R,2 (G) ≤ γF ,B,R (G) − |R|. By Claim 3.1, R ∩ {r1 , r2 , . . . , rk } = ∅; therefore D∗ − R = {r1 , r2 , . . . , rk } and |D∗ | − |R| = |D∗ − R|. Thus k = |{b1 , b2 , . . . , bk }| = |S ∗ | ≤ αF ,B,R,2 (G) ≤ γF ,B,R (G) − |R| ≤ |D∗ | − |R| = |D∗ − R| = |{r1 , r2 , . . . , rk }| = k. The three inequalities must be equalities. Hence D∗ is a minimum (F , B, R)-dominating set and S ∗ is a maximum (F , B, R)-2stable set of G. Moreover, αF ,B,R,2 (G) = γF ,B,R (G) − |R|.
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We now analyze the time complexity of Algorithm 1. Recall that we use the adjacency-list representation of graphs. The most time-consuming steps in Algorithm 1 are lines 3, 9 and 11; it is not difficult to see that all the other steps can be done in O(|V | + |E |) time. Line 3 takes O(|V | + |E |) time because: |R| ≤ |V | and |N(R)| ≤ 2|E |. Line 9 takes O(|V | + |E |) time because: once the vertices and edges of G have been input, a scan of the adjacency-lists suffices to find vi∗ for every vi and it takes O(|V | + |E |) time. Note that several vertices in G may have the same vi∗ (for example, in Fig. 3, vi∗ = v9 for all i ∈ {7, 8, 9}). Thus we can use a mark array to indicate whether N(vi∗ ) is visited or not and only when N(vi∗ ) is visited for the first time, we relabel all the bound vertices in N(vi∗ ). Based on such an implementation, line 11 can be done in O(|V | + |E |) time. From the above, Algorithm 1 takes O(|V | + |E |) time. □ 4. Mutual transferability for connected strongly chordal graphs The purpose of this section is to prove mutual transferability among (F , B, R)-dominating sets in connected strongly chordal graphs G. The approach of the proof uses the same spirit in Section 3 for the primal–dual method. First a lemma. Lemma 4.1. Let G be a connected strongly chordal graph with a strong elimination ordering v1 , v2 , . . . , vn . If i < n, then the largest-indexed vertex vi∗ in N [vi ] has i∗ > i. Proof. Choose a shortest vi -vn path vi1 vi2 . . . vim with vi1 = vi and vim = vn . Suppose there is some vir with ir < i, say ir is the smallest index with this property. Then 1 < r < m and vir −1 , vir +1 ∈ Nir [vir ], which gives that vir −1 vir +1 ∈ E(G) by the property of the strong elimination ordering. This leads to a shorter vi -vn path, a contradiction. Hence i2 > i and so i∗ > i. □ Theorem 4.2. For any connected strongly chordal graph G, if D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |, ∗ then D1 → D2 . Proof. Choose a strong elimination ordering v1 , v2 , . . . , vn of G. We shall prove a more general statement as follows by a back induction. ∗
(Si ) If D1 and D2 agree on {v1 , v2 , . . . , vi } (i.e., for each 1 ≤ j ≤ i, vj ∈ D1 iff vj ∈ D2 ), then D1 → D2 . Notice that the theorem is just (S0 ). (Sn ) is true as in this case D1 = D2 . Suppose 1 ≤ i ≤ n and (Si ) is true. To prove (Si−1 ) being true, suppose D1 and D2 agree on {v1 , v2 , . . . , vi−1 }. Suppose D1 has c1 copies of vi and D2 has c2 copies of vi . We may assume that c1 > c2 since nothing needs to be done if c1 = c2 and the case c2 > c1 can be proven similarly. Also, c1 > c2 ˆi = B ∩ (N [vi ] − N [D1 − {vi }]) for easy writing. Intuitively, N ˆi denotes the set of private bound neighbors implies i < n. Set N of vi with respect to D1 , i.e., bound vertices that only can be dominated by vi . Define j to be the minimum index such that ˆi , where j may or may not exist. There are two cases. vj ∈ N Case 1: c2 > 0 or j does not exist or j ≥ i. By Lemma 4.1, the largest-indexed vertex vi∗ in N [vi ] has i∗ > i. Moving c1 –c2 copies of vi of D1 to vi∗ results a set D′1 . If c2 > 0, then c2 copies of vi of D1 stay at vi of D′1 , and so D′1 is an (F , B, R)-dominating set ∗
with D1 → D′1 . If j does not exist, then vi has no private bound neighbors. If j ≥ i, then all the private bound neighbors of vi with respect to D1 are in Ni [vi ] ⊆ Ni [vi∗ ]. Also c2 = 0 gives vi ∈ F ∪ B. Therefore D′1 is an (F , B, R)-dominating set ∗
with D1 → D′1 . Case 2: c2 = 0 and j < i. Since c2 = 0 and D1 and D2 agree on {v1 , v2 , . . . , vi−1 }, D2 must dominate vj by using a vertex vj′ . Notice that j′ > i since vj is a private bound neighbor of vi and D1 and D2 agree on {v1 , v2 , . . . , vi−1 }. Moving c1 copies of vi of D1 to vj′ results a set D′1 . All the private bound neighbors of vi with respect to D1 are in Nj [vi ] ⊆ Nj [vj′ ]. Also ∗
c2 = 0 gives vi ∈ F ∪ B. Therefore D′1 is an (F , B, R)-dominating set with D1 → D′1 . ∗
∗
In any case, D′1 and D2 agree on {v1 , v2 , . . . , vi }, by the induction hypothesis, D′1 → D2 and so D1 → D2 . (See Fig. 4 for an illustration of the proof.) □ By letting B = V (G) and F = R = ∅, we have the following corollary, which improves the first main result in [14]. Corollary 4.3. If G is a connected strongly chordal graph in which D1 and D2 are two dominating sets with |D1 | = |D2 |, then ∗ D1 → D2 . 5. Mutual transferability for cactus graphs Before going further, we define cactus graphs and describe the relationship between cactus graphs and trees. A cutvertex in a graph is a vertex whose removal increases the number of connected components. A block in a graph is a maximal connected subgraph of the graph having no cut-vertices. A cactus graph is a connected graph in which each block is either an edge or a cycle. Thus a tree is a cactus graph in which each block is an edge. A cactus graph is sometimes called a cactus tree and any two cycles in it have at most one vertex in common. In [10], Das mentioned that cactus graphs have many applications because these graphs can be used to model physical setting (especially in telecommunications and local area
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∗
Fig. 4. The process of D1 → D2 , where D1 = {v2 , v4 , v7 , v7 , v8 } and D2 = {v3 , v6 , v7 , v9 , v9 }, in the connected strongly chordal graph shown in Fig. 3(a), which has F = ∅, B = {v1 , v2 , . . . , v9 }, R = ∅, and strong elimination ordering v1 , v2 , . . . , v9 . (a) (S1 ) H⇒ (S2 ). (b) (S3 ) H⇒ (S4 ). (c) (S4 ) H⇒ (S5 ). (d) ∗
(S6 ) H⇒ (S7 ). (e) (S7 ) H⇒ (S8 ). (f) (S9 ) holds; the process D1 → D2 now completes.
networks) where a tree would be inappropriate. In [16], Hedetniemi et al. proposed a linear-time algorithm, called MCACTUS, for finding a minimum (F , B, R)-dominating set in a cactus graph, which in turn uses an algorithm called MCYCLE to find a minimum (F , B, R)-dominating set in a cycle or an edge. In what follows, let G = (V , E) be a cactus graph and V be the disjoint union of F , B, and R. A block of G is called an end-block if it contains exactly one cut-vertex. We call an end-block of G a cycle end-block if it is a cycle. The rest of this section is organized as follows. Section 5.1 deals with the case that G is exactly a cycle. Section 5.2 deals with the case that G has a cycle end-block. Section 5.3 handles the general case. 5.1. G is a cycle This subsection deals with the case that G is a cycle. For clarity, denote G by C . The following algorithm, called Algorithm 2, will convert any minimum (F , B, R)-dominating set of C into a minimum (F , B, R)-dominating set D∗ of C that is suitable for performing mutual transferability. See Fig. 5 for an illustration. One claim is made. Claim 5.1. d(bi , bj ) > 2 (and hence d(ri , rj ) > 2) for any 1 ≤ i < j ≤ k except when {i, j} = {1, k}. Proof. Since D is a minimum (F , B, R)-dominating set of C , the set (S ∩ BI ) − T cannot be empty. Otherwise, we can find an (F , B, R)-dominating set of C having a cardinality smaller than |D|, a contradiction. Therefore b1 , b2 , . . . , bk and r1 , r2 , . . . , rk exist. By lines 7 and 8 in Algorithm 2, for each 1 ≤ t ≤ k, vertex rt is the successor of vertex bt (in clockwise ordering). Thus d(bi , bj ) ≤ 2 implies that bj ∈ N [ri ] and hence bj ̸ ∈ (S ∩ BI ) − T , which contradicts with bj ∈ (S ∩ BI ) − T (see line 6 in Algorithm 2). Since ri is the successor of bi and rj is the successor of bj (in clockwise ordering), we also have d(ri , rj ) > 2. □ Theorem 5.2. For any cycle C of n vertices and any minimum (F , B, R)-dominating set D of C , Algorithm 2 converts D into a ∗ minimum (F , B, R)-dominating set D∗ of C that satisfies Claim 5.1 in O(n) time. Moreover, D∗ = R ∪ {r1 , r2 , . . . , rk } and D → D∗ .
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Algorithm 2 Input: (1) A cycle C = (V , E) with its vertices v1 , v2 , . . . , vn listed in clockwise ordering and V being the disjoint union of F , B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. (2) A minimum (F , B, R)-dominating set D = R ∪ {vi1 , vi2 , . . . , vik } of C , where i1 < i2 < · · · < ik . Output: A minimum (F , B, R)-dominating set D∗ of C that satisfies Claim 5.1. ∗ 1: D ← R; 2: BI ← B − N(R); //BI = {important bound vertices, i.e., bound vertices not dominated by R} 3: for j = 1 to k do //replace vij with the vertex rj found below 4: S ← {vij −1 , vij , vij +1 }; //we assume vij −1 = vn for vij = v1 and vij +1 = v1 for vij = vn 5: if (j = 1) then T ← ∅; else T ← N [rj−1 ]; end if 6: vf ← the first-encountered bound vertex in (S ∩ BI ) − T under clockwise ordering vij −1 , vij , vij +1 ; 7: bj ← vf ; 8: rj ← vf +1 ; //b1 , b2 , . . . , bk and r1 , r2 , . . . , rk are used in proving correctness and mutually transferability 9: D∗ ← D∗ ∪ {rj }; 10: end for
Fig. 5. An illustration of Algorithm 2. R = {v5 }, the boxed node. (a) D = R ∪ {v2 , v8 , v10 }. (b) D∗ = R ∪ {v2 , v8 , v11 }.
Proof. Since D is a minimum (F , B, R)-dominating set of C , the set (S ∩ BI ) − T is non-empty. Therefore, during the execution of Algorithm 2, we can derive b1 , b2 , . . . , bk and r1 , r2 , . . . , rk . So D∗ = R ∪ {r1 , r2 , . . . , rk } is obvious. Since all the bound vertices dominated by D are dominated by D∗ and |D∗ | = |D|, it follows that D∗ is a minimum (F , B, R)-dominating set of C . Thus Algorithm 2 converts D into a minimum (F , B, R)-dominating set D∗ of C that satisfies Claim 5.1. We now analyze its time complexity. The most time-consuming steps in Algorithm 2 are lines 2, 5 and 6; all the other steps can be done in O(n) time. Since C is a cycle, each vertex has only two neighbors. Thus lines 2 and 5 takes O(n) time. Moreover, each vij has at most three bound vertices in its (S ∩ BI ) − T ; thus line 6 takes O(n) time. From the above, Algorithm 2 takes O(n) time. For mutual transferability, by moving vij to rj for j = 1, 2, . . . , k; each of the above moves is safe since every vertex in BI ∩ N [vi1 ] ∗
is dominated by r1 and every vertex in BI ∩ N [vij ], j ≥ 2, is dominated by rj−1 or rj . Thus D → D∗ .
□
Now we prove mutual transferability for (F , B, R)-dominating sets of a cycle. Theorem 5.3. Let C be a cycle and let its vertices be the disjoint union of F , B, and R. Let D1 be any (F , B, R)-dominating set of ∗ C , D∗ be the minimum (F , B, R)-dominating set of C produced by Algorithm 2, and ℓ = |D1 | − |D∗ |. Then D1 ∪ {1 · extra} → ∗ D ∪ {(ℓ + 1) · extra}. Proof. Let v1 , v2 , . . . , vn be the vertices of C listed in clockwise ordering around C . Note that by Algorithm 2 we obtain b1 , b2 , . . . , bk and r1 , r2 , . . . , rk with D∗ = R ∪ {r1 , . . . , rk }. Without loss of generality, we assume that an extra element in D1 ∪ {1 · extra} is located at v1 and assume that (ℓ + 1) extra elements in D∗ ∪ {(ℓ + 1) · extra} are also located at v1 . Let ∗ Y = D∗ ∪ {(ℓ + 1) · extra}. The process of performing D1 ∪ {1 · extra} → Y and its correctness are listed below. (See Fig. 6.) ◀ Standard Process for a Cycle ▶
1: For i = 1 to k − 1, let (ri , ri+1 ] denote the set of vertices in C between ri (exclusive) and ri+1 (inclusive) in a clockwise manner. Let (rk , r1 ) denote the set of vertices in C between rk and r1 (both exclusive) in a clockwise manner. (Take Fig. 6(b) as an example. Then, k = 3 and we have (r1 , r2 ] = (v2 , v8 ] = {v3 , v4 , v5 , v6 , v7 , v8 }, (r2 , r3 ] = (v8 , v11 ] = {v9 , v10 , v11 }, and (r3 , r1 ) = (v11 , v2 ) = {v1 }.) (1)
2: Move an extra element in D1 ∪ {1 · extra} to r1 . Such a move is safe since the moved element is extra. Let D1 be the resultant (F , B, R)-dominating set.
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∗
Fig. 6. The process of D1 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra}, where D1 = {v1 , v4 , v5 , v8 , v10 }, D∗ = {v2 , v5 , v8 , v11 }, ℓ = 1, and R = {v5 } (1) (2) (3) (the boxed node). (a) D1 ∪ {1 · extra}. (b) D1 = {v1 , v2 , v4 , v5 , v8 , v10 }. (c) D1 = {v1 , v2 , v5 , v8 , v8 , v10 }. (d) D1 = {v1 , v2 , v5 , v8 , v11 , v11 }. (e) (4)
D1 = {v2 , v5 , v8 , v11 , v1 , v1 } = D∗ ∪ {2 · extra}. (i)
3: For i = 1 to k − 1 do the following. For all of the elements in D1 that are located at ri , keep one of them at ri and move all of (i) the others to ri+1 . Then, move all of the elements in D1 that are located in (ri , ri+1 ] to ri+1 . By Claim 5.1, d(bi , bi+1 ) > 2; (i) thus N [bi+1 ] ⊆ (ri , ri+1 ], meaning that at least one element in D1 is located in (ri , ri+1 ] and therefore after the moves, at least one element is located at ri+1 . Notice that each of the above moves is safe since all the bound vertices in (ri , ri+1 ] (i+1) are dominated by {ri , ri+1 } ∪ R. Let the resultant (F , B, R)-dominating set be D1 . (k)
4: For all of the elements in D1 that are located at rk , keep one of them at rk and move all of the others to v1 . Then, move all (k) of the elements in D1 that are located in (rk , r1 ) to v1 . Each of the above moves is safe since all the bound vertices in (k+1) (k+1) is exactly Y . The set D1 (rk , r1 ) are dominated by {rk , v1 , r1 } ∪ R. Let the resultant (F , B, R)-dominating set be D1 and one extra element in Y is located at v1 . □ 5.2. G has a cycle end-block Let C be a cycle end-block of G and x be its unique cut-vertex. Following reference [16], we use CF and CR to denote C with x being relabeled as free and required, respectively, and DCF , DC and DCR to denote the minimum (F , B, R)-dominating set obtained by MCYCLE(CF ), MCYCLE(C ) and MCYCLE(CR ), respectively. Reference [16] says that |DCF | ≤ |DC | ≤ |DCR | < |DCF | + 1. There is a typo and the statement should be |DCF | ≤ |DC | ≤ |DCR | ≤ |DCF | + 1. Let D1 be any (F , B, R)-dominating set of G. Let C be any cycle end-block of G and x its unique cut-vertex. We use C − x to denote the graph obtained by deleting x from C . Set AF = D1 ∩ V (C − x)
and A = D1 ∩ V (C )
for easy writing. One lemma is needed. Lemma 5.4. Let G be a cactus graph with a cycle end-block C , x be the unique cut-vertex in C , and D1 be any (F , B, R)-dominating set of G. Then:
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• If |DCF | < |DC |, then |AF | ≥ |DCF |. • If |DCF | = |DC | < |DCR |, then |A | ≥ |DC |. • If |DCF | = |DC | = |DCR |, then |A | ≥ |DCR |. Proof. If |DCF | < |DC |, then AF ∪ {x} is an (F , B, R)-dominating set of C and DC is a minimum (F , B, R)-dominating set of C ; thus we have |AF | + 1 ≥ |DC | > |DCF |, that is, |AF | ≥ |DCF |. If |DCF | = |DC |, then A is an (F , B, R)-dominating set of CF and DCF is a minimum (F , B, R)-dominating set of CF ; thus we have |A | ≥ |DCF | = |DC |. Moreover, if |DCF | = |DC | = |DCR |, then we have |A | ≥ |DCF | = |DCR |. □ By Lemma 5.4, when |DCF | < |DC |, we have |AF | ≥ |DCF |; thus we define:
• D1 (DCF ) = (D1 − AF ) ∪ DCF ∪ {x, x, . . . , x}
(total number of x is |AF |−|DCF |).
As an example, if DCF = {v3 , v6 , v9 } and |AF | = 5, then DCF ∪ {x, x, . . . , x} is {v3 , v6 , v9 , x, x}. By Lemma 5.4, when |DCF | = |DC |, we have |A | ≥ |DC | or |A | ≥ |DCR |; thus we define:
• D1 (DC ) = (D1 − A ) ∪ DC ∪ {x, x, . . . , x} (total number of x is |A | − |DC |) x is |A | − |DC |) • D1 (DCR ) = (D1 − A ) ∪ DCR ∪ {x, x, . . . , x} (total number of x is |A | − |DCR |). The followings are clear:
• DCF ⊆ D1 (DCF ), DC ⊆ D1 (DC ), DCR ⊆ D1 (DCR ) • |D1 (DCF )| = |D1 (DC )| = |D1 (DCR )| = |D1 | • Each of D1 (DCF ), D1 (DC ), and D1 (DCR ) is an (F , B, R)-dominating set of G. Lemma 5.5. Let G be a cactus graph with a cycle end-block C , x be the unique cut-vertex in C , and D1 be any (F , B, R)-dominating set of G. ∗
• If |DCF | < |DC |, then D1 ∪ {1 · extra} → D1 (DCF ) ∪ {1 · extra}. ∗ • If |DCF | = |DC | < |DCR |, then D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra}. ∗ • If |DCF | = |DC | = |DCR |, then D1 ∪ {1 · extra} → D1 (DCR ) ∪ {1 · extra}. Proof. Let v1 (= x), v2 , . . . , vn be the vertices of C listed in clockwise ordering around C . Without loss of generality, we assume that an extra element in D1 ∪ {1 · extra} is located at v1 .
• If |DCF | < |DC |, let Z = CF , X = DCF and Y = D1 (DCF ) ∪ {1 · extra}. • If |DCF | = |DC | < |DCR |, let Z = C , X = DC and Y = D1 (DC ) ∪ {1 · extra}. • If |DCF | = |DC | = |DCR |, let Z = CR , X = DCR and Y = D1 (DCR ) ∪ {1 · extra}. ∗
Let the first input of Algorithm 2 be Z and the second input be X , and run Algorithm 2. Then, the process of D1 ∪{1 · extra} → Y and its correctness follows from Standard Process for a Cycle in Theorem 5.3. □ 5.3. The general case Now we deal with the general case of cactus graphs. ∗
Theorem 5.6. For any cactus graph G, if D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |, then D1 ∪{1 · extra} → D2 ∪ {1 · extra}. Proof. If G is a tree, then this theorem follows from Theorem 4.2 (actually, no extra element is needed). In what follows, we assume that G is not a tree and is of order at least three. We prove by induction on the number m of blocks of G. If m = 1, ∗ ∗ then G is a cycle. By Theorem 5.3, D1 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra} and D2 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra} ∗ both are true, where D be the minimum (F , B, R)-dominating set of G produced by Algorithm 2 and ℓ = |D1 | − |D∗ |. ∗ ∗ Notice that D2 ∪ {1 · extra} → D∗ ∪ {(ℓ + 1) · extra} implies D∗ ∪ {(ℓ + 1) · extra} → D2 ∪ {1 · extra}. Combining ∗ ∗ ∗ ∗ ∗ D1 ∪ {1 · extra} → D ∪ {(ℓ + 1) · extra} and D ∪ {(ℓ + 1) · extra} → D2 ∪ {1 · extra} obtains D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, meaning that the theorem holds when m = 1. In the following, let m ≥ 2 and assume that the theorem holds for cactus graphs having m − 1 blocks. There are two cases. Case 1: G has an end-block that is an edge. Then, let u be the vertex of degree 1 in this end-block and let x be the unique neighbor of u. Then, x must be a cut-vertex in G. Depending on the label of u, there are three subcases. 1.1: u is required. Then, at least one element in D1 must be u. Similarly, at least one element in D2 must be u. Let G′ = G − u and relabel x as free if its label in G is bound. Let D′′1 (respectively, D′′2 ) be the multiset obtained from D1 (respectively, D2 ) by keeping only one element at u and moving all of the other elements that are
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located at u to x. Let D′1 and D′2 be the multiset obtained from D′′1 and D′′2 by deleting the unique u in D′′1 and D′′2 , respectively. 1.2: u is free. Then, let G′ = G − u. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (respectively, D2 ) by moving every element located at u to x. 1.3: u is bound. Then, since u is bound, at least one element in D1 must be u or x. Similarly, at least one element in D2 must be u or x. Let G′ = G − u and relabel x as required. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (respectively, D2 ) by moving every element located at u to x. In any of the subcases 1.1, 1.2, and 1.3, D′1 and D′2 are (F , B, R)-dominating sets of G′ with |D′1 | = |D′2 |. Since u and x are adjacent, all the moves used in obtaining D′1 and D′2 are safe. Moreover, G′ is a cactus graph with m − 1 blocks. ∗
∗
By the induction hypothesis, D′1 ∪ {1 · extra} → D′2 ∪ {1 · extra}. Thus D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. Case 2: Every end-block in G is a cycle. Then, pick up an arbitrary end-block C in G and let x be the unique cut-vertex in C . Depending on the label of x, there are three subcases. 2.1: x is required. First we execute MCYCLE(C ) to obtain DC ; then use DC to obtain D1 (DC ) and D2 (DC ). Next we run Algorithm 2 with C and DC as its inputs, and we follow Standard Process for a Cycle in Theorem 5.3 to obtain ∗ ∗ D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra} and D2 ∪ {1 · extra} → D2 (DC ) ∪ {1 · extra}. Let G′ = G − (C − x). Let D′1 ′ (respectively, D2 ) be the multiset obtained from D1 (DC ) (respectively, D2 (DC )) by deleting all of its elements appearing in C − x. 2.2: x is free. In this case, |DCF | = |DC | occurs. Therefore, we first execute MCYCLE(C ) and MCYCLE(CR ) to obtain DC and DCR ; then use DC and DCR to obtain D1 (DC ), D2 (DC ), D1 (DCR ), and D2 (DCR ). There are two subcases. ∗
2.2.1: |DCF | = |DC | < |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra} and ∗
D2 ∪ {1 · extra} → D2 (DC ) ∪ {1 · extra}. Let G′ = G − (C − x). Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DC ) (respectively, D2 (DC )) by deleting all of its elements appearing in C − x. ∗ 2.2.2: |DCF | = |DC | = |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DCR ) ∪ {1 · extra} and ∗
D2 ∪ {1 · extra} → D2 (DCR ) ∪ {1 · extra}. Let G′ = G − (C − x) and relabel x in G′ as required. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DCR ) (respectively, D2 (DCR )) by deleting all of its elements appearing in C − x. 2.3: x is bound. Then, we first execute MCYCLE(CF ), MCYCLE(C ) and MCYCLE(CR ) to obtain DCF , DC and DCR ; then use DCF , DC and DCR to obtain D1 (DCF ), D2 (DCF ), D1 (DC ), D2 (DC ), D1 (DCR ), and D2 (DCR ). ∗
∗
2.3.1: |DCF | < |DC |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DCF ) ∪ {1 · extra} and D2 ∪ {1 · extra} → D2 (DCF ) ∪ {1 · extra}. Let G′ = G − (C − x). Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DCF ) (respectively, D2 (DCF )) by deleting all of its elements appearing in C − x. ∗
2.3.2: |DCF | = |DC | < |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DC ) ∪ {1 · extra} and ∗
D2 ∪{1 · extra} → D2 (DC ) ∪{1 · extra}. Let G′ = G − (C − x) and relabel x in G′ as free. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DC ) (respectively, D2 (DC )) by deleting all of its elements appearing in C − x. ∗ 2.3.3: |DCF | = |DC | = |DCR |. Then, by Lemma 5.5, we have D1 ∪ {1 · extra} → D1 (DCR ) ∪ {1 · extra} and ∗
D2 ∪ {1 · extra} → D2 (DCR ) ∪ {1 · extra}. Let G′ = G − (C − x) and relabel x in G′ as required. Let D′1 (respectively, D′2 ) be the multiset obtained from D1 (DCR ) (respectively, D2 (DCR )) by deleting all of its elements appearing in C − x. In any of the subcases 2.1, 2.2.1, 2.2.2, 2.3.1, 2.3.2, and 2.3.3, D′1 and D′2 are (F , B, R)-dominating sets of G′ with |D′1 | = ∗
|D′2 |. Moreover, G′ is a cactus graph with m − 1 blocks. By the induction hypothesis, D′1 ∪ {1 · extra} → D′2 ∪ {1 · extra}. ∗ Thus D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. We now complete the proof. □ By letting B = V (G) and F = R = ∅, we have the following corollary. ∗
Corollary 5.7. If G is a cactus graph in which D1 and D2 are two dominating sets with |D1 | = |D2 |, then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}. 6. Concluding remarks In this paper, we have proved any strongly chordal graph has the strong duality equality for (F , B, R)-domination. We ∗ ∗ have also shown that if G is connected strongly chordal then D1 → D2 , and if G is a cactus graph then D1 ∪ {1 · extra} → D2 ∪ {1 · extra}, where D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |. A preliminary version of this paper was in [7].
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For any connected graph G, we may try to determine the least number ℓ(G) of extra elements required to guarantee ∗ mutually transferability. That is, if D1 and D2 are two (F , B, R)-dominating sets of G with |D1 | = |D2 |, then D1 ∪ {ℓ · extra} → D2 ∪ {ℓ · extra} for ℓ ≥ ℓ(G). For example, ℓ(G) = 0 when G is a connected strongly chordal graph, and ℓ(G) = 1 when G is a cactus graph. We conjecture that any connected graph G has ℓ(G) ≤ 1. In fact, for cycles Cn , we yield the following partial results: if all vertices are labeled as bound (or equivalently, ordinary domination), then ℓ(Cn ) = 1 if n = 3r + 6 and ℓ(Cn ) = 0 otherwise; if there is a vertex labeled as required, then ℓ(Cn ) = 0; and if no vertex is labeled as required and there are adjacent vertices labeled as free, then ℓ(Cn ) = 0. Acknowledgments The authors would like to thank Prof. Gerard Jennhwa Chang and Prof. Sheng-Chyang Liaw for their valuable comments that greatly improved the presentation of this paper. The authors would also like to express their deepest gratitude to the anonymous reviewers for their constructive comments and time spent to analyze this paper. References [1] R.P. Anstee, M. Farber, Characterizations of totally balanced matrices, J. Algorithms 5 (2) (1984) 215–230. [2] K.S. Booth, J.H. Johnson, Dominating sets in chordal graphs, SIAM J. Comput. 11 (1) (1982) 191–199. [3] G.J. Chang, k-domination and graph covering problems (Ph.D thesis), School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1982. [4] G.J. Chang, Algorithmic aspects of domination in graphs, in: D.-Z. Du, P.M. Pardalos (Eds.), Handbook of Combinatorial Optimization, vol. 3, 1998, pp. 339–405. [5] G.J. Chang, G.L. Nemhauser, The k-domination and k-stability problems in sun-free chordal graphs, SIAM J. Algebr. Discrete Methods 5 (3) (1984) 332–345. [6] L. Chen, C. Lu, Z. Zeng, A linear-time algorithm for paired-domination problem in strongly chordal graphs, Inform. Process. Lett. 110 (1) (2009) 20–23. [7] K.T. Chu, Mutual transferability of dominating sets in strongly chordal graphs and cactus graphs, (Master thesis), National Tsing Hua University, Hsinchu, Taiwan, 2018. [8] E.J. Cockayne, S. Goodman, S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1) (1975) 41–44. [9] D.G. Corneil, L.K. Stewart, Dominating sets in perfect graphs, Discrete Math. 86 (1–3) (1990) 145–164. [10] K. Das, Some algorithms on cactus graphs, Ann. Pure Appl. Math. 2 (2) (2012) 114–128. [11] M. Farber, Applications of linear programming duality to problems involving independence and domination (Ph.D Thesis), Rutgers University, New Brunswick, NJ, 1982, also issued as Technical Report 81-13, Computing Science Department, Simon Fraser University, 1981. [12] M. Farber, Characterizations of strongly chordal graphs, Discrete Math. 43 (2–3) (1983) 173–189. [13] M. Farber, Domination, independent domination and duality in strongly chordal graphs, Discrete Appl. Math. 7 (2) (1984) 115–130. [14] S. Fujita, A tight bound on the number of mobile servers to guarantee transferability among dominating configurations, Discrete Appl. Math. 158 (8) (2010) 913–920. [15] P.A. Golovach, P. Heggernes, N. Lindzey, R.M. McConnell, V.F. dos Santos, J.P. Spinrad, J.L. Szwarcfiter, On recognition of threshold tolenrance graphs and their complements, Discrete Appl. Math. 216 (1) (2017) 171–180. [16] S.T. Hedetniemi, R. Laskar, J. Pfaff, A linear algorithm for finding a minimum dominating set in a cactus, Discrete Appl. Math. 13 (2–3) (1986) 287–292. [17] A. Kolen, Solving covering problems and the uncapacitated plant location problem on trees, Eur. J. Oper. Res. 12 (3) (1983) 266–278. [18] R. Laskar, J. Pfaff, S.M. Hedetniemi, S.T. Hedetniemi, On the algorithmic complexity of total domination, SIAM J. Algebr. Discrete Methods 5 (3) (1984) 420–425. [19] A. Lubiw, Doubly lexical orderings of matrices, SIAM J. Comput. 16 (5) (1987) 854–879. [20] R. Nevries, C. Rosenke, Characterizing and computing the structure of clique intersections in strongly chordal graphs, Discrete Appl. Math. 181 (2015) 221–234. [21] R. Paige, R.E. Tarjan, Three partition refinement algorithms, SIAM J. Comput. 16 (6) (1987) 973–989. [22] J. Sawada, J.P. Spinrad, From a simple elimination ordering to a strong elimination ordering in linear time, Inform. Process. Lett. 86 (6) (2003) 299–302.