An improved algorithm to construct edge-independent spanning trees in augmented cubes

Discrete Applied Mathematics xxx (xxxx) xxx

Contents lists available at ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

An improved algorithm to construct edge-independent spanning trees in augmented cubes ∗

Baolei Cheng a,b , Jianxi Fan a,b , , Cheng-Kuan Lin c , Yan Wang a , Guijuan Wang a a

School of Computer Science and Technology, Soochow University, Suzhou 215006, China Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks, Najing, Jiangsu 21000, China c College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China b

article

info

Article history: Received 21 January 2019 Received in revised form 2 August 2019 Accepted 25 September 2019 Available online xxxx Keywords: Augmented cube Edge-independent spanning trees Height Fault-tolerant broadcasting

a b s t r a c t Edge-Independent spanning trees (EISTs) can be used in IP fast rerouting to guarantee recovery from link failures, as well as in data broadcasting in networks to enhance fault-tolerance, bandwidth, and security. The n-dimensional augmented cube AQn is an important node-symmetric variant of the n-dimensional hypercube Qn . Recently, Wang et al. proposed a method to construct 2n − 1 EISTs in AQn (Wang et al., 2017). However, the height of the tallest EIST is no less than n+2. Since height is an important performance indicator of the EISTs, in this paper, we further study the existence and construction of EISTs in augmented cubes with lower heights. We first propose a constructive algorithm to construct 2n − 1 trees rooted at any node in AQn . We then prove the 2n − 1 trees obtained by the algorithm are 2n − 1 EISTs with the maximal height n, and the time complexity of the algorithm is O(N log N), where N = 2n is the number of nodes in AQn . © 2019 Elsevier B.V. All rights reserved.

1. Introduction When studying the communication strategies in interconnection networks, they are usually abstracted as graphs [1,4, 39,40]. Given a simple graph G, its node set and edge set are represented by V (G) and E(G), respectively. The union of a graph G1 and a graph G2 is represented by G1 ∪ G2 , with the node set V (G1 ) ∪ V (G2 ) and the edge set E(G1 ) ∪ E(G2 ). If W is a nonempty subset of V (G), then the subgraph of G induced by W is denoted by G[W ]. A graph G1 is said to be isomorphic to graph G2 , if there is a bijection ψ : V (G1 ) → V (G2 ) such that (ψ (u), ψ (v )) ∈ E(G2 ) if and only if (u, v ) ∈ E(G1 ). Let x, y ∈ V (G). A path R from x to y is denoted as x-y path. We use V (R) and E(R) to denote the node set and the edge set in R, respectively. Two x-y paths P and Q are edge-disjoint if E(P) ∩ E(Q ) = ∅. We use H(T ) to denote the height of a tree T . Two spanning trees T1 and T2 rooted at a node u in graph G are edge-independent if the u-v path in T1 and the u-v path in T2 are edge-disjoint for each v ∈ V (G)\{u}. A set of spanning trees in G are edge-independent if they are pairwisely edge-independent. In this paper, we will simply use EISTs to denote edge-independent spanning trees unless otherwise specified. EISTs have various applications in reliable communication protocols [1,2], the multi-node broadcasting [4], one-to-all broadcasting [38], fault-tolerant broadcasting, and secure message distribution [1], IP fast rerouting [15,19]. According to fault-tolerant broadcasting protocol [2,23,26], suppose that there exist n EISTs rooted at the same source node u in graph G. If a message m is required to be broadcasted to every other node from the root node u, each node in G will obtain n ∗ Corresponding author at: School of Computer Science and Technology, Soochow University, Suzhou 215006, China. E-mail addresses: [email protected] (B. Cheng), [email protected] (J. Fan), [email protected] (C.-K. Lin), [email protected] (Y. Wang), [email protected] (G. Wang). https://doi.org/10.1016/j.dam.2019.09.021 0166-218X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

2

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

copies of m by sending m along the n EISTs. For any fault-free source node, this scheme can tolerate up to n − 1 faulty edges. Thus, recently the problem of constructing multiple EISTs in a given graph has received much attention. There exists an edge conjecture on the existence of EISTs in graphs [26,48]: Conjecture 1.1. If a graph G is n-edge-connected (n ≥ 1), then there are n EISTs rooted at an arbitrary node in G. But until now, Conjecture 1.1 only has been solved for any n-edge-connected graph with n ≤ 4 [1,11,13,18,20]. At the same time, the conjecture has been proven to hold for several restricted classes of graphs, such as planar graphs [24,25,33,34], product graphs [35], chordal rings [27,46], deBruijn and Kautz graphs [17], even networks [28], odd networks [29], crossed cubes [6,7,9], Mo¨ bius cubes [5,8], locally twisted cubes [22,30], twisted cubes [41,43], folded hypercubes [45], augmented cubes [42], folded hyper-stars [44], multidimensional torus networks [37], recursive circulant graphs [47], etc. Researchers focus on not only the proofs of existence of EISTs, but also the trees with high performance such as low heights, where low heights are useful to design broadcasting algorithms with high efficiency. To reduce the maximum height of a set of EISTs is a difficult thing, until now, only several results have been obtained [9,36,46]. In certain circumstances, to reduce the maximum height of a set of EISTs means to reduce the number of EISTs [9]. The n-dimensional augmented cube, AQn , is both (2n − 1)-edge-connected and (2n − 1)-node-connected, which has received wide interest from researchers [3,10,12,14,16,21,31,32,42]. Since the edge-connectivity is 2n − 1, the optimal number of EISTs rooted at an arbitrary node in AQn is 2n − 1. Although Wang et al. have solved Conjecture 1.1 for AQn [42], the maximum height of the set of EISTs is more than n and the heights of the set of EISTs are not equal. This motivated us to ask the following question: Do there exist 2n − 1 EISTs rooted at any node in AQn with lower height? In this paper, we study the existence and construction of 2n − 1 EISTs rooted at an arbitrary node with low height in AQn . Our major contributions are as follows: 1. For any integer n ≥ 1 and for an arbitrary node u in AQn , an O(N log N) algorithm, called AIST, is developed to construct 2n − 1 EISTs T1 , T2 , . . . , T2n−1 rooted at node u in AQn , where N = 2n is the number of nodes in AQn . 2. We prove that the 2n − 1 EISTs rooted at an arbitrary node obtained by Algorithm AIST are all of the maximal height n. The rest of this paper is organized as follows. Section 2 provides the preliminaries. Section 3 gives a constructive proof that there exist 2n − 1 EISTs rooted at an arbitrary node with height n in AQn and presents an O(N log N) algorithm. In Section 4, we conclude the paper. 2. Preliminaries A binary string x of length n can be written as xn−1 xn−2 . . . x0 , where xn−1 is the most significant bit and x0 is the least significant bit. We say xi ∈ {0, 1} to be the ith bit of x for integer i with 0 ≤ i ≤ n − 1. The complement of xi will be denoted by xi (0 = 1 and 1 = 0). Given a set W ⊆ {0, 1}n , if there exists a string s ∈ {0, 1}m such that s is a prefix of z for all z ∈ W for some integer m with 1 ≤ m ≤ n − 1, then s is called a common prefix of all the binary strings in W . We adopt the definition of AQn as follows. Definition 2.1 ([12]). For n ≥ 1, the augmented cube of dimension n, AQn , has 2n nodes, each labeled by an n-bit binary string an−1 an−2 . . . a0 . We define AQ1 = K2 . For n ≥ 2, AQn is obtained by taking two copies of the augmented cube AQn−1 , denoted by AQn0−1 and AQn1−1 , and adding 2 × 2n−1 edges between the two as follows: Let V (AQn0−1 ) = {0an−2 an−3 . . . a0 |an−2 an−3 . . . a0 ∈ V (AQn−1 )} and V (AQn1−1 ) = {1bn−2 bn−3 . . . b0 |bn−2 bn−3 . . . b0 ∈ V (AQn−1 )}. A node a = 0an−2 an−3 . . . a0 of AQn0−1 is joined to a node b = 1bn−2 bn−3 . . . b0 of AQn1−1 if and only if either (i) ai = bi for every i, 0 ≤ i ≤ n − 2; in this case, (a, b) is called a hypercube edge, or (ii) ai = (bi ) for every i, 0 ≤ i ≤ n − 2; in this case, (a, b) is called a complement edge. Clearly, Qn is a proper subgraph of AQn . The following lemma, originally presented in [12], describes how to check if a given pair of nodes are adjacent in AQn . At this time, the lemma can be considered as the non-recursive definition of augmented cubes. Lemma 2.1 ([12]). For all n ≥ 1, (un−1 un−2 . . . u0 , vn−1 vn−2 . . . v0 ) is an edge if and only if there exists an integer ℓ, 0 ≤ ℓ ≤ n − 1 such that either (1) uℓ = vℓ and ui = vi , for every i, i ̸ = l, or (2) ui = vi for ℓ + 1 ≤ i ≤ n − 1 and ui = vi for 0 ≤ i ≤ ℓ. When either condition 1 or condition 2 of Lemma 2.1 holds, we say that u and v have a leftmost differing bit at position ℓ. When two adjacent nodes u and v have a leftmost differing bit at position d and ud−1 ud−2 . . . u0 = vd−1 vd−2 . . . v0 , we say that v is the d-neighbor of u and the edge (u, v ) is an edge of dimension d. We let γ (u, d) denote the d-neighbor of u. When two adjacent nodes x and y have a leftmost differing bit at position d and xd−1 xd−2 . . . x0 = yd−1 yd−2 . . . y0 , we say Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

3

Fig. 1. (a) AQ1 ; (b) AQ2 ; (c) AQ3 ; (d) AQ4 .

that the edge (x, y) is a complement edge of dimension d and y is the d-complement-neighbor of x. We let ξ (x, d) denote the neighbor of x in the complement edge. Furthermore, for W ⊆ V (AQn ), γ (W , d) = {γ (w, d)|w ∈ W } denotes the set of the d-neighbors of all nodes in W . For ′ ′ ′ ′ ′ ′ W ⊆ V (AQn ), ξ (W , d) = {ξ (w , d)|w ∈ W } denotes the set of the d-complement-neighbors of all nodes in W . In Fig. 1, (a), (b), (c), and (d) demonstrate AQ1 , AQ2 , AQ3 , and AQ4 , respectively. AQ1 and AQ2 are complete graphs. AQ3 is composed of two AQ2 and AQ4 is composed of two AQ3 . For integer m with 1 ≤ m ≤ n − 1 and string s ∈ {0, 1}m , AQn (s) is defined as the subgraph of AQn induced by the set of nodes whose addresses have the common prefix s. Lemma 2.2. AQn (s) is isomorphic to AQn−|s| , where |s| denotes the length of a string s. As suggested by Lemma 2.2, for integer n with n ≥ 2, if u = un−1 un−2 . . . u0 is an arbitrary node in AQn , then u belongs to the subgraph of AQn : AQn (un−1 un−2 . . . un−i ) for integer i with 1 ≤ i ≤ n − 1. Similar to AQn , we can u i−1 un−i−1 divide AQn (un−1 un−2 . . . un−i ) into two subgraphs AQn (un−1 un−2 . . . un−i )nn−−i− 1 and AQn (un−1 un−2 . . . un−i )n−i−1 . For example, 0 1 AQ4 (1) is composed of AQ4 (1)2 and AQ4 (1)2 . In Fig. 1(d), the graph surrounded by dotted line is AQ4 (1)12 . Let T be a tree rooted at node u in AQn . We use ancestor(v, T ) to represent the set of all the nodes in u-v path except v itself in T , and ancestorEdge(v, T ) to represent the set of all the edges in u-v path in T . The height of T is the maximum distance of the path from u to any other node in T . For example, in Fig. 2(b), ancestorEdge(15, T1 )={(0, 1), (1, 9), (9, 13), (13, 15)} and H(T1 ) = 4. 3. Construction of edge-independent spanning trees with low height 3.1. Analysis of the heights of EISTs Although Wang et al. do not present the heights of EISTs in [42], we can easily obtain the value, see Fig. 2(a) (decimal notation). For n = 3, the heights of the 5 EISTs T1 , T2 , . . . , T5 are 4, 4, 4, 4, 3, respectively. The tallest one is 4 and the lowest one is 3. For n = 4, the heights of the 7 EISTs T1 , T2 , . . . , T7 are 5, 4, 4, 5, 4, 4, 6, respectively. The tallest one is 6 and the lowest one is 4. But we can present another set of EISTs rooted at 0 with the maximum height 4 for n = 4. See Fig. 2(b) (decimal notation). For n ≥ 5, the 2n − 1 EISTs rooted at 0 in AQn obtained by [42] are constructed based on the 2n − 3 EISTs rooted at 0 in AQn−1 with recursive mode. Take T7 for example, in AQ4 , H(T7 ) = 6. During every iteration, the height of T7 increases by 1. Thus, in AQn , the height of T7 is n − 4 + 6 = n + 2, which implies that the maximum height of the 2n − 1 EISTs obtained by [42] is no less than n + 2. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

4

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

Fig. 2. (a) 7 EISTs rooted at 0 in AQ4 ; (b) 7 EISTs rooted at 0 in AQ4 .

3.2. Edge-independent spanning trees in AQ2 and AQ3 Now we discuss a new construction scheme of EISTs rooted at node 0 in AQn for n=1, 2, 3. Let T1 be a tree rooted at 0 in AQ1 with V (T1 ) = {0, 1} and E(T1 ) = {(0, 1)}. The three trees T1 , T2 , and T3 rooted at 00 in AQ2 are as follows: V (T1 ) = V (T2 ) = V (T3 ) = {00, 01, 10, 11}. E(T1 ) = {(00, 01), (01, 10), (01, 11)}. E(T2 ) = {(00, 10), (10, 01), (10, 11)}. E(T3 ) = {(00, 11), (11, 01), (11, 10)}. It is easy to verify that the three paths between the root node 00 and any other node in {01, 10, 11} are edge-disjoint. Thus, T1 , T2 , and T3 are three EISTs rooted at 00 in AQ2 . Since AQ20 is obtained by appending a prefix 0 to every node in AQ2 , we can obtain the three EISTs TA,1 , TA,2 , and TA,3 rooted at 000 in AQ20 by appending a prefix 0 to every node in the three EISTs T1 , T2 , and T3 rooted at 00 in AQ2 . Now we discuss the steps to construct 2 × 3 − 1 = 5 trees rooted at 000 in AQ3 based on the 3 EISTs rooted at node 00 in AQ2 . Step 1: Construct three isomorphic EISTs TB,1 , TB,2 , and TB,3 rooted at 100 in AQ21 by complementing the 2nd bit of every node in TA,1 , TA,2 , and TA,3 . Step 2: For trees T1 and T2 , connect node 001 with 101 and node 010 with 110, respectively. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

5

Fig. 3. EISTs of T1 , T2 , . . . , T5 rooted at node 000 in AQ3 .

Step 3: For tree T3 , connect nodes 001, 010, 011 with nodes 110, 101, 100, respectively. Furthermore, connect node 011 with node 111 and disconnect edges (100, 111), (111, 101), and (111, 110). Step 4: For tree T4 , firstly, we construct the subtree T4′ with V (T4′ ) = {100, 101, 110, 111} and E(T4′ ) = {(100, 101), (100, 110), (100, 111)}. Note that the three edges in E(T4′ ) come from TB,1 , TB,2 , and TB,3 . Then, we connect nodes 100, 101, and 110 with 011, 010, and 001, respectively. Finally, connect node 100 with 000. Step 5: For tree T5 , firstly, we construct the subtree T5′ with V (T5′ ) = {100, 101, 110, 111} and E(T5 ) = {(111, 100), (111, 101), (111, 110)}, which comes from TB,3 . Then, we connect nodes 101, 110, and 111 with nodes 001, 010, and 011, respectively. Finally, connect node 111 with 000. It is easy to verify that the five paths between the root node 000 and any other node in {001, 010, 011, 100, 101, 110, 111} are edge-disjoint. Thus, T1 , T2 , . . . , T5 are five EISTs rooted at 000 in AQ3 . Fig. 3 shows the construction of 5 EISTs rooted at 000 in AQ3 . Note that there also exists other set of EISTs T1′ , T2′ , . . . , T5′ similar to T1 , T2 , . . . , T5 , respectively. See Fig. 4 for example. Since AQn (s) is defined as the subgraph of AQn induced by the set of nodes whose addresses have the common prefix s, we can easily find the induced subgraph AQn (un−1 un−2 . . . u1 ), AQn (un−1 un−2 . . . u2 ), . . . , AQn (un−1 ), which is isomorphic to AQ1 , AQ2 , . . . , AQn−1 , respectively. We will discuss constructing the EISTs rooted at u in high dimensional augmented cube based on the EISTs in low dimension augmented cube in the next subsection. 3.3. Construction of edge-independent spanning trees in AQn In this section, by the results in the last subsection, we first give some preliminary results about the construction of 2n − 1 EISTs rooted at an arbitrary node in AQn . Then, we provide a recursive algorithm to construct 2n − 1 EISTs rooted at an arbitrary node in AQn , analyze the time complexity, and discuss the heights of EISTs. In the following, we always use u = un−1 un−2 . . . u0 to denote an arbitrarily chosen root node in AQn . Now, we present an algorithm, called AIOT, as follows. uk−1 For integer n with n ≥ k ≥ 2, given a spanning tree TA rooted at node u in AQn (un−1 un−2 . . . uk )k− 1 , we will obtain a u

k−1 tree TB rooted at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 and isomorphic to TA by Algorithm AIOT. By Algorithm AIOT, if n > k, all nodes in TA have the prefix un−1 un−2 . . . uk−1 and all nodes in TB have the prefix un−1 un−2 . . . uk uk−1 . Otherwise, all nodes in TA have the prefix un−1 = uk−1 and all nodes in TB have the prefix un−1 = uk−1 . In Fig. 5, (a) and (b) demonstrate the constructions of tree TB in AQ31 with respect to the root nodes 0000 and 0001 of tree TA in AQ30 ,⨁ respectively. Given that is the XOR operation, we have the following lemma.

u

k−1 Lemma 3.1. For a node v in AQn (un−1 un−2 . . . uk )k− 1 ,v

bit of v .

⨁

00 ...0 1 00 ...0 = γ (v, k − 1), which is complementing the (k − 1)th     n−k

k−1

Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

6

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

Fig. 4. EISTs of T1′ , T2′ , . . . , T5′ rooted at node 000 in AQ3 .

Fig. 5. The constructions of tree TB in AQ31 with respect to tree TA rooted at nodes (a) 0000 and (b) 0001 in AQ30 , respectively.

Algorithm 1:

AIOT. u

k−1 Input: A spanning tree TA rooted at node u = un−1 un−2 . . . u0 in AQn (un−1 un−2 . . . uk )k− 1 , where n ≥ k ≥ 2. Output: A tree TB . Begin ⨁ ⨁ 1: V ← {v 00 ...0 1 00 ...0 | v ∈ V (TA )}; /*Here, is XOR operation*/    

n−k

2:

E ← {(x

⨁

k−1

00 ...0 1 00 ...0, y     n−k

Return TB = (V , E); End

k−1

⨁

...0 | (x, y) ∈ E(TA )}; 00 ...0 1 00     n−k

k−1

3:

Proof. Let v = vn−1 vn−2 . . . v0 . We have

v

⨁

00 ...0 1 00 ...0 = vn−1 vn−2 . . . v0     n−k

k−1

⨁

...0 1 00 ...0 = vn−1 vn−2 . . . vk vk−1 vk−2 vk−3 . . . v0 . 00   n−k

k−1

That is, complement the (k − 1)th bit of v . Since γ (v, k − 1) = vn−1 vn−2 . . . vk vk−1 vk−2 vk−3 . . . v0 , the lemma holds. □ Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

7

By Algorithm AIOT, we have the following lemma. Lemma 3.2. For integers n and k with n ≥ k ≥ 2, let (TA , u) be the input and let TB be the output of Algorithm AIOT. Then TB uk−1 is a spanning tree rooted at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 isomorphic to TA . Proof. By Algorithm AIOT and Lemma 3.1, since TA is a spanning tree rooted at node u = un−1 un−2 . . . u0 in AQn (un−1 un−2 uk−1 . . . uk )k− 1 which has the common prefix un−1 un−2 . . . uk uk−1 , TB is obtained by complementing the (k−1)th bit of each node in TA . TB is isomorphic to TA . The difference between TA and TB is that the common prefix changed from un−1 un−2 . . . uk uk−1 uk−1 into un−1 un−2 . . . uk uk−1 . By Definition 2.1, we can verify that TB is a spanning tree in AQn (un−1 un−2 . . . uk )k− 1 , where n ≥ k ≥ 2. □ By Algorithm AIOT and Lemma 3.2, it is easy to verify that the following lemma. Lemma 3.3. For integers n and k with n ≥ k ≥ 2, let (TA , TA′ ) be the input and let (TB , TB′ ) be the output of Algorithm AIOT, respectively. If TA and TA′ are two EISTs rooted at node u, then TB and TB′ are two EISTs rooted at node γ (u, k − 1) in u

k−1 AQn (un−1 un−2 . . . uk )k− 1 .

For the convenience of constructing 2k − 1 EISTs rooted at node u in AQn (un−1 un−2 . . . uk ), we give the following two definitions and two corollaries. Definition 3.1. For integers n and k with n ≥ k ≥ 2, suppose that TA is a spanning tree rooted at node u in uk−1 AQn (un−1 un−2 . . . uk )k− 1 . If there exists a nonempty set W ⊆ V (TA )\{u} such that for v ∈ W , ancestor(v, TA )\{u} ⊆ W , then we call W an extendable node set (ENS) with respect to the spanning tree TA . The following result is clear. u

k−1 Corollary 3.1. Let TA be a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk )k− 1 . If there exists an ENS W with respect to TA , then TA [W ∪ {u}] is a tree rooted at u.

Definition 3.2. For integers n and k with n ≥ k ≥ 2, suppose TA,1 , TA,2 , . . . , TA,2k−3 are 2k − 3 EISTs rooted at node u in uk−1 AQn (un−1 un−2 . . . uk )k− 1 and W1 , W2 , . . . , W2k−3 are ENSs with respect to TA,1 , TA,2 , . . . , TA,2k−3 , respectively. If the following two conditions hold: (1) For two integers i and j with 1 ≤ i < j ≤ 2k − 3, Wi ∩ Wj = ∅, and ⋃2k−3 uk−1 (2) i=1 Wi ∪ {u} = V (AQn (un−1 un−2 . . . uk )k− 1 ), uk−1 then we call {{u}, W1 , W2 , . . . , W2k−3 } an extendable node set partition (ENSP) on V (AQn (un−1 un−2 . . . uk )k− 1 ). For example, Fig. 3 demonstrates the case n = k = 3. {{000}, {001}, {010}, {011}} is an ENSP on V (AQ20 ), where {001}, {010}, {011} are ENSs with respect to TA,1 , TA,2 , TA,3 , respectively, rooted at 000 in AQ20 . Notation 3.1. We use η to denote the mapping from TA to TB defined by Algorithm AIOT. By Algorithm AIOT, we have the following lemma. Lemma 3.4. Let W be a set with W ⊆ V (TA ), η(W ) = γ (W , k − 1). Based on the above results, we have the following corollary. u

k−1 Corollary 3.2. For integer n with n ≥ 2, let a spanning tree TA rooted at node u in AQn (un−1 un−2 . . . uk )k− 1 be the input and let TB be the tree obtained by Algorithm AIOT. If W is an ENS with respect to TA , then TB [γ (W ∪ {u}, k − 1)] is a tree rooted at uk−1 node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 .

u

k−1 Proof. By Corollary 3.1, TA [W ∪ {u}] is a tree rooted at u in AQn (un−1 un−2 . . . uk )k− 1 . By Lemma 3.2, TB is a spanning tree

uk−1 uk )k− 1

rooted at γ (u, k − 1) in AQn (un−1 un−2 . . . and isomorphic to TA . By Lemma 3.4, η(W ) = γ (W , k − 1). Thus η(W ∪ {u}) = η(W ) ∪ {η(u)} = γ (W ∪ {u}, k − 1). uk−1 As a result, TB [η(W ∪ {u})] = TB [γ (W ∪ {u}, k − 1)] is a tree rooted at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 . □ For example, TB,1 [γ (W1 ∪ {000}, 2)], TB,2 [γ (W2 ∪ {000}, 2)], and TB,3 [γ (W3 ∪ {000}, 2)] in AQ21 shown in Fig. 3 are three trees rooted at node γ (000, 2) = 100. Given a node u in AQn , the key point is as follows: u How to construct 2n − 1 EISTs rooted at node u in AQn based on the 2n − 3 EISTs rooted at node u in AQn n−1 and let the height of each EIST equal to the dimension n. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

8

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

Fig. 6. T1 , T2 , . . . , T7 are not EISTs rooted at node 0000 in AQ4 .

As a consequence, we will give an algorithm, called AIT, as follows. Now we show that the input of Algorithm AIT should satisfy additional conditions to construct EISTs, that is how to choose ENSP is also a question. For example, in Fig. 6, TA,1 , TA,2 ,. . . , TA,5 are five EISTs rooted at 0000 in AQ40 , and W1 , W2 ,. . . , W5 such that W1 = {0001, 0101}, W2 = {0010, 0110}, W3 = {0011}, W4 = {0100}, and W5 = {0111} are ENSs with respect to TA,1 , TA,2 , TA,3 , TA,4 , and TA,5 , respectively, and {{0000}, W1 , W2 , . . . , W5 } is an ENSP on V (AQ40 ). By Algorithm AIT, 7 trees T1 , T2 , . . . , T7 are constructed. We can verify that in T1 and T5 , the edge (0001,0101) both appears in the 0000-1110 path, then T1 , T2 , . . . , T7 shown in Fig. 6 do not form a set of EISTs rooted at node 0000. But if we let W1 = {0001}, W2 = {0010}, W3 = {0011}, W4 = {0100, 0101, 0110}, and W5 = {0111}. By Algorithm AIT, with a similar construction, we can verify the 7 trees shown in Fig. 7 form a set of EISTs rooted at node 0000. Based on the definition of EISTs in Section 1, we present an edge properties between a node v and the root node u in EISTs. Lemma 3.5. Let Ti and Tj (i ̸ = j) be two spanning trees rooted at node u in G. Ti and Tj are edge-independent if and only if for every node v in G with v ̸ = u, ancestorEdge (v, Ti ) ∩ ancestorEdge (v, Tj ) = ∅. Based on the definition of EISTs, Lemma 3.5 and Definition 2.1, we have the following lemma. Lemma 3.6. Suppose that AQk has 2k − 1 EISTs rooted at a node u. Then, the edges (u, γ (u, 0)), (u, γ (u, 1)), (u, ξ (u, 1)), (u, γ (u, 2)), (u, ξ (u, 2)), . . . , (u, γ (u, k − 1)), (u, ξ (u, k − 1)) can be used exactly once in the 2k − 1 EISTs. When an edge appears in a set of EISTs only once, we say it is a unique edge. It is easy to verify that the edges in the 2k − 1 EISTs shown in Lemma 3.6 are all unique edges. Now we add some restricted conditions to the input of Algorithm AIT, based on which, we prove that the 2k − 1 trees T1 , T2 , . . . , T2k−1 obtained by Algorithm AIT are 2k − 1 EISTs rooted at node u in AQn (un−1 un−2 . . . uk ) in Theorem 3.1. Theorem 3.1. For integer n ≥ k ≥ 3, let the inputs be 2k − 3 EISTs TA,1 , TA,2 , . . . , TA,2k−3 rooted at node u in AQn (un−1 uk−1 un−2 . . . uk )k− 1 and W1 , W2 ,. . . , W2k−3 such that W1 , W2 ,. . . , W2k−3 are ENSs with respect to TA,1 , TA,2 , . . . , TA,2k−3 , respectively, uk−1 and {{u}, W1 , W2 , . . . , W2k−3 } is an ENSP on V (AQn (un−1 un−2 . . . uk )k− 1 ). If the following conditions hold: Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

9

Fig. 7. EISTs of T1 , T2 , . . . , T7 rooted at node 0000 in AQ4 .

(i) (u, γ (u, 0)), (u, γ (u, 1)), (u, ξ (u, 1)), (u, γ (u, 2)), (u, ξ (u, 2)), . . . , (u, γ (u, k − 2)), (u, ξ (u, k − 2)) are the unique edges in TA,1 , TA,2 , . . . , TA,2k−3 , respectively. (ii) W1 = {γ (u, 0)}, W2j−1 = {ξ (u, j − 1)} for j=2, 3, . . . , k − 1, uj−1

W2j−2 = V (AQn (un−1 un−2 . . . uj )j−1 )\W2j−1 for j=2, 3, . . . , k − 1. /*Note that the selection of Wi is a key point for i = 1 to 2k − 3.*/ (iii) H(TA,i ) = k − 1 and H(TA,i [Wi ∪ {u}]) ≤ k − 2 for i = 1, 2, . . . , 2k − 3. (iv) E(TA,i [Wi ∪ {u}]) ∩ E(TA,2k−3 ) = ∅ for i = 1, 2, . . . , 2k − 4. Then (i) T1 , T2 , . . . , T2k−1 obtained by Algorithm AIT are 2k − 1 EISTs rooted at node u in AQn (un−1 un−2 . . . uk ). (ii) H(T1 ) = H(T2 ) = · · · = H(T2k−1 ) = k. Proof. By Algorithm AIT and Lemma 3.2, 2k − 3 trees TB,1 , TB,2 , . . . , TB,2k−3 rooted at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk−1 uk )k− 1 are obtained by using Algorithm AIOT based on TA,1 , TA,2 , . . . , TA,2k−3 rooted at u. Clearly, H(TB,i ) = k − 1 for i = 1, 2, . . . , 2k − 3. By Lemma 3.2 and Corollary 3.2, and condition (i) in the hypothesis, we can easily verify that the following results hold: (i) (γ (u, k − 1), γ (γ (u, k − 1), 0)), (γ (u, k − 1), γ (γ (u, k − 1), 1)), (γ (u, k − 1), ξ (γ (u, k − 1), 1)), (γ (u, k − 1), γ (γ (u, k − 1), 2)), (γ (u, k − 1), ξ (γ (u, k − 1), 2)), . . . , (γ (u, k − 1), γ (γ (u, k − 1), k − 2)), (γ (u, k − 1), ξ (u, k − 2)) are the unique edges in TB,1 , TB,2 , . . . , TB,2k−3 , respectively. (ii) γ (W1 , k − 1) = γ ({γ (u, 0)}, k − 1), γ (W2j−1 , k − 1) = γ ({ξ (u, j − 1)}, k − 1), for j=2, 3, . . . , k − 1. u

γ (W2j−2 , k − 1) = γ (V (AQn (un−1 un−2 . . . uj )j−j−11 )\W2j−1 , k − 1), for j=2, 3, . . . , k − 1. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

10

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

Algorithm 2: AIT. u

k−1 Input: 2k − 3 EISTs TA,1 , TA,2 , . . . , TA,2k−3 rooted at node u in AQn (un−1 un−2 . . . uk )k− 1 and W1 , W2 ,. . . , W2k−3 such that W1 , W2 ,. . . , W2k−3 are ENSs with respect to TA,1 , TA,2 , . . . , TA,2k−3 , respectively, and {{u}, W1 , W2 , . . . , W2k−3 } is an ENSP on uk−1 V (AQn (un−1 un−2 . . . uk )k− 1 ), where n ≥ k ≥ 2. Output: 2k − 1 trees T1 , T2 , . . . , T2k−1 . Begin Step 1:

u

k−1 Construct 2k − 3 trees TB,1 , TB,2 , . . . , TB,2k−3 rooted at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 by using Algorithm AIOT based on TA,1 , TA,2 , . . . , TA,2k−3 rooted at u, respectively. Step 2: Construct T1 , T2 , . . . , T2k−4 in AQn (un−1 un−2 . . . uk ) as follows. 2: for i = 1 to 2k − 4 do 3: V (Ti ) ← V (AQn (un−1 un−2 . . . uk )) 4: E(Ti ) ← E(TA,i ) ∪ E(TB,i ) ∪ {(w, γ (w, k − 1))|w ∈ Wi }\E(TB,i [γ (Wi , k − 1)]) 5: end for Construct T2k−3 in AQn (un−1 un−2 . . . uk ) as follows. 6: V (T2k−3 ) ← V (AQn (un−1 un−2 . . . uk )) 7: E(T2k−3 ) ← E(TA,2k−3 ) ∪ {(w, ξ (w, k − 1))|w ∈ V (TA,2k−3 ) \{un−1 un−2 . . . u0 }} ∪{(ξ (u, k − 2), γ (ξ (u, k − 2), k − 1)} Step 3: /*Construct tree T2k−2 as follows:*/

1:

2k−3

8:

TB,2k−2 ←

⋃

TB,i [γ (Wi ∪ {u}, k − 1)]

i=1

V (T2k−2 ) ← V (AQn (un−1 un−2 . . . uk )) E(T2k−2 ) ← E(TB,2k−2 ) ∪ {(z , ξ (z , k − 1))|z ∈ V (TB,2k−2 )\{ξ (u, k − 1)}} 11: E(T2k−2 ) ← E(T2k−2 ) ∪ {(u, γ (u, k − 1))} Step 4: /* Construct tree T2k−1 as follows:*/ 12: TB,2k−1 ← TB,2k−3 13: V (T2k−1 ) ← V (AQn (un−1 un−2 . . . uk )) 14: E(T2k−1 ) ← E(TB,2k−1 ) ∪ {(z , γ (z , k − 1))|z ∈ V (TB,2k−1 )\{γ (u, k − 1)}} 15: E(T2k−1 ) ← E(T2k−1 ) ∪ {(u, ξ (u, k − 1))} 16: Return T1 , T2 , . . . , T2k−1 End 9:

10:

/*Note that the selection of Wi is a key point for i = 1 to 2k − 3.*/ (iii) H(TB,i ) = k − 1 and H(TB,i [γ (Wi ∪ {u}, k1 )]) ≤ k − 2 for i = 1, 2, . . . , 2k − 3. (iv) E(TB,i [γ (Wi ∪ {u}, k − 1)]) ∩ E(TB,2k−3 ) = ∅ for i = 1, 2, . . . , 2k − 4. Clearly, T1 , T2 , . . . , and T2k−3 are 2k − 3 spanning trees rooted at node u in AQn (un−1 un−2 . . . uk ). By Algorithm AIT, we have H(Ti ) = H(TA,i ) + 1 = k − 1 + 1 = k for k = 1, 2, . . . , 2k − 3. In what follows, we will first prove that T2k−2 is a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk ) and H(T2k−2 ) = k, then prove T2k−1 is a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk ) and H(T2k−1 ) = k. Next, we will prove that T1 , T2 , . . . , T2k−1 are edge-independent. (1) Firstly, we prove that T2k−2 is a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk ) and H(T2k−2 ) = k. uk−1 Recalling that TB,1 , TB,2 , . . . , TB,2k−3 are 2k − 3 EISTs rooted at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 . By ⋃2k−3 ⋃ uk−1 2k−3 Definition 3.2, we have i=1 Wi ∪ {u} = V (AQn (un−1 un−2 . . . uk )k− ). Furthermore, we have γ (W ∪ { u }, k − 1) = i 1 i=1 u

k−1 V (AQn (un−1 un−2 . . . uk )k− 1 ) and TB,2k−2 =

⋃2k−3

⋃2k−3 i=1

TB,i [γ (Wi ∪ {u}, k − 1)]. Hence, u

k−1 V (TB,2k−2 ) = i=1 γ (Wi ∪ {u}, k − 1) = V (AQn (un−1 un−2 . . . uk )k− 1 ). Since TB,i [γ (Wi ∪ {u}, k − 1)] is a tree rooted at γ (u, k − 1), we can easily verify that TB,2k−2 is a tree rooted uk−1 at node γ (u, k − 1) in AQn (un−1 un−2 . . . uk )k− 1 . As a result, TB,2k−2 is a spanning tree rooted at node γ (u, k − 1) in

u

k−1 AQn (un−1 un−2 . . . uk )k− 1 . Since V (T2k−2 ) = V (AQn (un−1 un−2 . . . uk )) and E(T2k−2 ) = E(TB,2k−2 ) ∪{(z , ξ (z , k − 1))|z ∈ V (TB,2k−2 )}, T2k−2 is a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk ). By Algorithm AIT, H(T2k−2 ) = max{H(TB,1 [γ (W1 ∪{u}, k − 1)]), H(TB,2 [γ (W2 ∪ {u}, k − 1)]), . . . , H(TB,2k−3 [γ (W2k−3 ∪ {u}, k − 1)])} + 2 = k − 2 + 2 = k. Secondly, we prove that T2k−1 is a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk ). By Algorithm AIT, TB,2k−1 = TB,2k−3 . By Lemma 3.2, TB,2k−1 is a spanning tree rooted at γ (u, k − 1) in AQn (un−1 un−2 . . . uk−1 uk )k− 1 . By Algorithm AIT, V (T2k−1 ) = V (AQn (un−1 un−2 . . . uk )), E(T2k−1 ) = E(TB,2k−1 ) ∪ {(z , γ (z , k − 1))|z ∈ V (TB,2k−1 )\{γ (u, k − 1)}}, and E(T2k−1 ) = E(T2k−1 ) ∪ {(u, ξ (u, k − 1))}. We can verify that T2k−1 is a spanning tree rooted at node u in AQn (un−1 un−2 . . . uk ) and H(T2k−1 ) = H(TB,2k−1 ) + 1 = k − 1 + 1 = k.

Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

11

(2) In what follows, we will first prove T1 , T2 , . . . , T2k−3 are edge-independent. We will then prove that T2k−1 , T2k−2 , Ti are edge-independent for integer i with 1 ≤ i ≤ 2k − 3. For integers i and j with 1 ≤ i < j ≤ 2k − 3 and for node v ∈ V (AQn (un−1 un−2 . . . uk )) with v ̸ = u, in what follows, let uk−1 uk−1 G = AQn (un−1 un−2 . . . uk ), G1 = AQn (un−1 un−2 . . . uk )k− 1 , and G2 = AQn (un−1 un−2 . . . uk )k−1 , we always define Xi = ancestorEdge(v, Ti ) ∩ E(G1 ), Yi = ancestorEdge(v, Ti ) ∩ E(G2 ), XYi = ancestorEdge(v, Ti ) ∩ E(G)\(E(G1 ) ∪ E(G2 )), Xj = ancestorEdge(v, Tj ) ∩ E(G1 ), Yj = ancestorEdge(v, Tj ) ∩ E(G2 ), XYj = ancestorEdge(v, Tj ) ∩ E(G)\(E(G1 ) ∪ E(G2 )) Clearly, Xi ∩ Yi = ∅, Xi ∩ XYi = ∅, Yi ∩ XYi = ∅, Xj ∩ Yj = ∅, Xj ∩ XYj = ∅, Yj ∩ XYj = ∅, Xi ∩ Yj = ∅, Xi ∩ XYj = ∅, Yi ∩ XYj = ∅, Xj ∩ Yi = ∅, Xj ∩ XYi = ∅, and Yj ∩ XYi = ∅. Thus, ancestorEdge(v, Ti ) ∩ ancestorEdge(v, Tj ) = (Xi ∪ Yi ∪ XYi ) ∩ (Xj ∪ Yj ∪ XYj )

= (Xi ∩ Xj ) ∪ (Yi ∩ Yj ) ∪ (XYi ∩ XYj )

(3.1)

Furthermore, we prove the following four claims (i), (ii), (iii), and (iv): (i) T1 , T2 , . . . , T2k−3 are edge-independent. For integers i and j with 1 ≤ i < j ≤ 2k − 3 and for node v ∈ V (AQn (un−1 un−2 . . . uk )) with v ̸ = u, we have the cases as follows: Case 1. v ∈ V (G1 ). Obviously, XYi = Yi = ∅ and XYj = Yj = ∅. Thus, XYi ∩ XYj = ∅.

(3.2)

Yi ∩ Yj = ∅.

(3.3)

Since TA,i and TA,j are edge-independent, by Lemma 3.5, Xi ∩ Xj = ∅.

(3.4)

By Eqs. (3.1), (3.2), (3.3), and (3.4), we have ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. Case 2. v ∈ V (G2 ). We have the following two cases. Case 2.1. 1 ≤ i < j ≤ 2k − 4. By Algorithm AIT, Xi ⊆ TA,i [Wi ∪ {u}] and Xj ⊆ TA,j [Wj ∪ {u}]. Since TA,i and TA,j are edge-independent, we have Xi ∩ Xj = ∅ by Lemma 3.3. By Definition 3.2, Wi ∩ Wj = ∅, which implies that XYi ∩ XYj = ∅. Furthermore, by Lemma 3.3 and Algorithm AIT, TB,i and TB,j are edge-independent, Yi ∩ Yj = ∅. Then, ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = (Xi ∩ Xj ) ∪ (Yi ∩ Yj ) ∪ (XYi ∩ XYj ) = ∅. By Lemma 3.5, Ti and Tj are edge-independent. Case 2.2. 1 ≤ i ≤ 2k − 4 and j = 2k − 3. By condition (iv) in the hypothesis, E(TA,i [Wi ∪ {u}]) ∩ E(TA,2k−3 ) = ∅ for i = 1, 2, . . . , 2k − 4, which implies that Xi ∩ X2k−3 = ∅. By Algorithm AIT, Y2k−3 = ∅, XY2k−3 = {(v, ξ (v, k − 1))} ∪{(u, ξ (u, k − 2))} and XYi ⊆ {(w, γ (w, k − 1))|w ∈ Wi }. Then, we have Yi ∩ Y2k−3 = ∅ and XYi ∩ XY2k−3 = ∅. Hence, Ti and Tj are edge-independent. (ii) T2k−2 and Ti are edge-independent, where 1 ≤ i ≤ 2k − 3. We have the following two cases. Case 1. v ∈ G1 . By Algorithm AIT, X2k−2 = ∅, Yi = ∅, and XYi = ∅, which implies that X2k−2 ∩ Xi = ∅, Y2k−2 ∩ Yi = ∅, and XYi ∩ XYj = ∅. Hence, ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. Case 2. v ∈ G2 . By Algorithm AIT, X2k−2 = ∅ and XY2k−2 = {(u, γ (u, k − 1))}. By Algorithm AIT, (u, γ (u, k − 1)) is the unique edge in T2k−2 , which implies that X2k−2 ∩ Xi = ∅ and XYi ∩ XYj = ∅. By Algorithm AIT, Yi contains the edges that come from TB,ℓ . We have the following two cases. Case 2.1. ℓ = i. Then, all edges in TB,i [γ (W ∪ {u}, k − 1)] are disconnected, which implies that Yi = ∅. We have ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. Case 2.2. ℓ ̸ = i. Since TB,i and TB,k are edge-independent, we have Yi ∩ Yj = ∅. Thus, ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. (iii) T2k−1 and Ti are edge-independent, where 1 ≤ i ≤ 2k − 3. We have the following two cases. Case 1. v ∈ G1 . By Algorithm AIT, X2k−1 = ∅, Yi = ∅, and XYi = ∅, which implies that X2k−1 ∩ Xi = ∅, Y2k−1 ∩ Yi = ∅, and XYi ∩ XY2k−1 = ∅. Hence, ancestorEdge(v, Ti )∩ ancestorEdge(v, T2k−1 ) = ∅. Case 2. v ∈ G2 . By Algorithm AIT, X2k−1 = ∅ and XY2k−1 = {(u, ξ (u, k − 1))}. By Algorithm AIT, (u, ξ (u, k − 1)) is the unique edge in T2k−1 , which implies that X2k−1 ∩ Xi = ∅ and XYi ∩ XY2k−1 = ∅. By Algorithm AIT, Yi contains the edges that come from TB,2k−3 . We have the following two cases. Case 2.1. i = 2k − 3. Then, all edges in TB,i [γ (W ∪ {u}, k − 1)] are disconnected, which implies that Yi = ∅. We have ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

12

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

Case 2.2. i ̸ = 2k − 3. Since TB,i and TB,2k−3 are edge-independent, we have Yi ∩ Yj = ∅. Thus, ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. (iv) T2k−2 and T2k−1 are edge-independent. We have the following two cases. Case 1. v ∈ G1 . By Algorithm AIT, X2k−2 = X2k−1 = ∅, XY2k−2 = {(v, ξ (v, k − 1))} ∪ {(u, γ (u, k − 1))}, and XY2k−1 = {(v, γ (v, k − 1))}∪{(u, ξ (u, k − 1))}. Then, XY2k−2 ∩ XY2k−1 = ∅. By Algorithm AIT, Y2k−1 contains the edges in the u-v path that come from TB,2k−3 , and Y2k−2 contains the edges in the u-v path that come from TB,ℓ , where 1 ≤ ℓ ≤ 2k − 4. Since TB,2k−3 and TB,ℓ are edge-independent, we have Y2k−2 ∩ Y2k−1 = ∅. Thus, ancestorEdge(v, Ti )∩ ancestorEdge(v, Tj ) = ∅. Case 2. v ∈ G2 . The proof is similar to Case 1. In summary, T1 , T2 , . . . , T2k−1 are edge-independent. □ Fig. 7 demonstrates an ENSP {{0000}, {0001}, {0010}, {0011}, {0100,0101,0110}, {0111}} on V (AQ30 ), where {0001}, {0010}, {0011}, {0100,0101,0110}, and {0111} are ENSs with respect to TA,1 , TA,2 , TA,3 , TA,4 , and TA,5 rooted at 0000, respectively. In what follows, we will give an integrated algorithm, called AIST, to construct 2n − 1 EISTs rooted at node u in AQn with n ≥ 2. Given a node u = un−1 un−2 . . . u0 in AQn , we will construct the EISTs recursively. The base is the three EISTs rooted at u in AQn (un−1 un−2 ...u2 ). Based on the above three EISTs rooted at u in AQn (un−1 un−2 ...u2 ), the five EISTs rooted at u in AQn (un−1 un−2 ...u3 ) will be constructed. By recursive construction, based on the 2n − 3 EISTs rooted at u in AQn (un−1 ), the 2n − 1 EISTs rooted at u in AQn will be constructed. In Algorithm AIST, we first design two member variables k and n. Then, we will provide a rule to select an ENSP (see line 3 and lines 11–12 in Algorithm AIST). Algorithm 3: AIST(k, un−1 un−2 . . . u0 ). Input: An integer n with n ≥ k ≥ 2 and u = un−1 un−2 . . . u0 . Output: 2k − 1 trees T1 , T2 , . . . , T2k−1 rooted at node un−1 un−2 . . . u0 in AQn (un−1 un−2 . . . uk ), and 2k − 1 node sets W1 , W2 , . . . , W2k−1 . Begin Step 1: 1: if (k = 2) then 2: for i = 1 to 3 do 3: W1 ← {γ (u, 0)}, W2 ← {γ (u, 1)}, and W3 ← {ξ (u, 1)} 4: end for 5: return three trees T1 , T2 , T3 with V (T1 ), V (T2 ), V (T3 ) ← V (AQn (un−1 un−2 . . . u2 )) and E(T1 ) ← {(u, γ (u, 0)), (γ (u, 0), ξ (γ (u, 0), 1)), (γ (u, 0), γ (γ (u, 0), 1))}, E(T2 ) ← {(u, γ (u, 1)), (γ (u, 1), ξ (γ (u, 1), 1)), (γ (u, 1), γ (γ (u, 1), 0))}, and E(T3 ) ← {(u, ξ (u, 1)), (ξ (u, 1), γ (ξ (u, 1), 1)), (ξ (u, 1), γ (ξ (u, 1), 0)}. 6: else 7: call AIST(k − 1, un−2 un−3 . . . u0 ) 8: end if Step 2: 9: Let TA,1 , TA,2 , . . . , TA,2k−3 denote the 2k − 3 EISTs rooted at u in AQn (un−1 un−2 . . . uk−1 ). 10: Construct T1 , T2 , . . . , and T2k−1 rooted at node u in AQn (un−1 un−2 . . . uk ) by using Algorithm AIT based on TA,1 , TA,2 , . . . , TA,2k−3 and W1 , W2 ,. . . , W2k−3 such that W1 , W2 ,. . . , W2k−3 are ENSs with respect to TA,1 , TA,2 , . . . , TA,2k−3 , respectively, uk−1 and {{u}, W1 , W2 , . . . , W2k−1 } is an ENSP on V (AQn (un−1 un−2 . . . uk )k− 1 ). Step 3: 11: W2k−1 ← {ξ (u, k − 1)} /*W1 , W3 , W5 , ..., W2k−3 , W2k−1 contain only one node, respectively.*/ u

k−1 W2k−2 ← V (AQn (un−1 un−2 . . . uk )k− 1 )\W2k−1 ) /*There are 21 − 1 = 1, 22 − 1 = 3, . . . , 2k−1 − 1 nodes in W2 , W4 , W6 , ..., W2k−2 , respectively.*/ 13: Return T1 , T2 , . . . , T2k−1 End

12:

Theorem 3.2. Let k ≤ n. For integer n ≥ 2 and for an arbitrary node u = un−1 un−2 . . . u0 in AQn (un−1 un−2 . . . uk ), the following results hold: (i) T1 , T2 , . . . , T2k−1 obtained by Algorithm AIST are 2k − 1 EISTs rooted at node u in AQn (un−1 un−2 . . . uk ). (ii) The heights of T1 , T2 , . . . , T2k−1 are all equal to k. Proof. We prove the theorem by induction on k. Base: We first prove that the theorem holds for k = 2, 3, 4. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

13

(a) By Definition 2.1, there are only four nodes u = un−1 un−2 . . . u0 , un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 in AQn (un−1 un−2 . . . u2 ). By Algorithm AIST, we have V (T1 ) = V (T2 ) = V (T3 ) = {un−1 un−2 . . . u0 , un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 }. Furthermore, E(T1 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u1 u0 ), (un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u2 u1 u0 )}. E(T2 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 )}. E(T3 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 )}. W1 = {γ (u, 0)} = {un−1 un−2 . . . u1 u0 }. W2 = {γ (u, 1)} = {un−1 un−2 . . . u2 u1 u0 }. W3 = {ξ (u, 1)} = {un−1 un−2 . . . u2 u1 u0 }. It is easy to verify that T1 , T2 , and T3 are 3 EISTs rooted at u in AQn (un−1 un−2 . . . u2 ) and H(T1 ) = H(T2 ) = H(T3 ) = 2. u Let T1 , T2 , and T3 denote the 3 EISTs TA,1 , TA,2 , and TA,3 rooted at u in AQn (un−1 un−2 . . . u3 )22 , respectively. Observing that Algorithm AIST calls Algorithm AIT to construct T1 , T2 , . . . , T5 rooted at u, we have V (T1 ) = V (T2 ) = V (T3 ) = V (T4 ) = V (T5 ) = V (AQn (un−1 un−2 . . . u3 )). E(T1 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u1 u0 ), (un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2

. . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 )}, E(T2 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 )}, E(T3 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 )}, E(T4 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2

. . . u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u1 u0 )}, E(T5 ) = {(un−1 un−2 . . . u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u2 u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2

. . . u1 u0 ), (un−1 un−2 . . . u3 u2 u1 u0 , un−1 un−2 . . . u3 u2 u1 u0 )}. It is easy to verify that T1 , T2 , . . . , T5 are 5 EISTs rooted at u in AQn (un−1 un−2 . . . u3 ), whose heights are all equal to 3. See Fig. 3 for the five EISTs rooted at 000. Therefore, the theorem holds for n = 3. u Let T1 , T2 , . . . , T5 denote the 5 EISTs TA,1 , TA,2 , . . . , TA,5 rooted at u in AQn (un−1 un−2 . . . u4 )33 , respectively. By Algorithm AIST, we further have (i) (u, γ (u, 0)), (u, γ (u, 1)), (u, ξ (u, 1)), (u, γ (u, 2)), (u, ξ (u, 2)) are the unique edges in TA,1 , TA,2 , . . . , TA,5 , respectively. (ii) W1 = {γ (u, 0)}, u W3 = {ξ (u, 1)}, W2 = V (AQn (un−1 un−2 . . . u2 )11 )\W3 , u W5 = {ξ (u, 2)}, W4 = V (AQn (un−1 un−2 . . . u3 )22 )\W5 , (iii) H(TA,i ) = 3 and max{H(TA,1 [W1 ∪ {u}]), H(TA,2 [W2 ∪ {u}]), . . . , H(TA,5 [W5 ∪ {u}])} = 2 for i = 1, 2, . . . , 5. (iv) E(TA,i [Wi ∪ {u}]) ∩ E(TA,5 ) = ∅ for i = 1, 2, 3, 4. By Theorem 3.1, we have (i) T1 , T2 , . . . , T7 obtained by Algorithm AIT are 7 EISTs rooted at node u in AQn (un−1 un−2 . . . u4 ). (ii) H(T1 ) = H(T2 ) = · · · = H(T7 ) = 4. Therefore, the theorem holds for n = 4. u Let T1 , T2 , . . . , T7 denote the 7 EISTs TA,1 , TA,2 , . . . , TA,7 rooted at u in AQn (un−1 un−2 . . . u5 )44 , respectively. By Algorithm AIST, we can verify that the following conditions hold. (i) (u, γ (u, 0)), (u, γ (u, 1)), (u, ξ (u, 1)), (u, γ (u, 2)), (u, ξ (u, 2)), (u, γ (u, 3)), and (u, ξ (u, 3)) are the unique edges in TA,1 , TA,2 , . . . , TA,7 , respectively. Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

14

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx Table 1 The number of nodes in ENSs.

|W1 | |W2 | |W3 | |W4 | |W5 | |W6 | |W7 | |W8 | |W9 | |W10 | |W11 | |W12 | |W13 | |W14 | |W15 |

AQ2

AQ3

AQ4

AQ5

AQ6

AQ7

AQ8

1 1 1

1 1 1 3 1

1 1 1 3 1 7 1

1 1 1 3 1 7 1 15 1

1 1 1 3 1 7 1 15 1 31 1

1 1 1 3 1 7 1 15 1 31 1 63 1

1 1 1 3 1 7 1 15 1 31 1 1 1 127 1

(ii) W1 = {γ (u, 0)}, u W3 = {ξ (u, 1)}, W2 = V (AQn (un−1 un−2 . . . u2 )11 )\W3 , u W5 = {ξ (u, 2)}, W4 = V (AQn (un−1 un−2 . . . u3 )22 )\W5 , u W7 = {ξ (u, 3)}, W6 = V (AQn (un−1 un−2 . . . u4 )33 )\W7 . (iii) H(TA,i ) = 4 and max{H(TA,1 [W1 ∪ {u}]), H(TA,2 [W2 ∪ {u}]), . . . , H(TA,7 [W7 ∪ {u}])} = 3 for i = 1, 2, . . . , 7. (iv) E(TA,i [Wi ∪ {u}]) ∩ E(TA,7 ) = ∅ for i = 1, 2, . . . , 6. Induction: Suppose that the theorem holds for τ ≥ 4. That is, TA,1 , TA,2 , . . . , TA,2τ −1 are 2τ − 1 EISTs rooted at u in AQn (un−1 un−2 . . . uτ ) and the following conditions hold: (i) (u, γ (u, 0)), (u, γ (u, 1)), (u, ξ (u, 1)), (u, γ (u, 2)), (u, ξ (u, 2)), . . . , (u, γ (u, τ − 1)), (u, ξ (u, τ − 1)) are the unique edges in TA,1 , TA,2 , . . . , TA,2τ −1 , respectively. (ii) W1 = {γ (u, 0)}, W2j−1 = {ξ (u, j − 1)} for j=2, 3, . . . , τ . uj−1

W2j−2 = V (AQn (un−1 un−2 . . . uj )j−1 )\W2j−1 for j=2, 3, . . . , τ . (iii) H(TA,i ) = τ and max{H(TA,1 [W1 ∪ {u}]), H(TA,2 [W2 ∪ {u}]), . . . , H(TA,2τ −1 [W2τ −1 ∪ {u}])} = τ − 1 for i = 1, 2, . . . , 2τ − 1. (iv) E(TA,i [Wi ∪ {u}]) ∩ E(TA,2τ −1 ) = ∅ for i = 1, 2, . . . , 2τ − 2. Then By Theorem 3.1, we have (i) T1 , T2 , . . . , T2τ +1 obtained by Algorithm AIT are 2τ + 1 EISTs rooted at node u in AQn (un−1 un−2 . . . uτ +1 ). (ii) H(T1 ) = H(T2 ) = · · · = H(T2τ +1 ) = τ + 1 and H(T1 [W2τ +1 ∪ {u}]) = τ . By Algorithm AIST, we further can verify that (i) (u, γ (u, 0)), (u, γ (u, 1)), (u, ξ (u, 1)), (u, γ (u, 2)), (ξ (u, 2)), . . . , (u, γ (u, τ )), (u, ξ (u, τ )) are the edges in TA,1 , TA,2 , . . . , TA,2τ +1 , respectively. (ii) W1 = {γ (u, 0)}, W2j−1 = {ξ (u, j − 1)} for j=2, 3, . . . , τ + 1. uj−1

W2j−2 = V (AQn (un−1 un−2 . . . uj )j−1 )\W2j−1 for j=2, 3, . . . , τ + 1. (iii) H(TA,i ) = τ + 1 and max{H(TA,1 [W1 ∪ {u}]), H(TA,2 [W2 ∪ {u}]), . . . , H(TA,2τ +1 [W2τ +1 ∪ {u}])} = τ for i = 1, 2, . . . , 2τ + 1. (iv) E(TA,i [Wi ∪ {u}]) ∩ E(TA,2τ +1 ) = ∅ for i = 1, 2, . . . , 2τ . As a consequence, we have proven that the theorem holds for n = τ + 1. □ For example, by Algorithm AIST, we can easily obtain the number of nodes in ENSs W1 , W2 , . . . , W2n−1 with respect to EISTs rooted at u in AQn for n = 2, 3, . . . , 8. See Table 1. The nodes in the ENSs (decimal notation) can also easily be obtained by Algorithm AIST. For example, let the root node be 0 and k = n = 8. We have W1 = {1}, W2 = {2}, W3 = {3}, W4 = {4, 5, 6}, W5 = {7}, W6 = {8, 9, . . . , 14}, W7 = {15}, W8 = {16, 17, . . . , 30}, W9 = {31}, W10 = {32, 33, . . . , 62}, W11 = {63}, W12 = {64, 65, . . . , 126}, W13 = {127}, W14 = {128, 129, . . . , 254}, W15 = {255}. Figs. 3 and 7 demonstrate the constructive procedures of 2n − 1 EISTs rooted at node 0 (decimal notation) in AQn by Algorithm AIST for n = 3, 4. Observing that the 2k − 1 EISTs rooted at u in AQn (un−1 un−2 . . . uk ) are constructed in the recursive mode based on the 2k − 3 EISTs rooted at u, it is not difficult to obtain that the time complexity of Algorithm AIST is O(N log N). Theorem 3.3. For integer n with n ≥ 2, Algorithm AIST obtains 2n − 1 EISTs rooted at an arbitrary node u in AQn in O(N log N) time, where N = 2n is the number of nodes in AQn . Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

15

4. Conclusion Although [42] proposed a method to construct 2n − 1 EISTs in AQn , the maximal height of the set of EISTs is bigger. In this paper, we study the construction of 2n − 1 EISTs rooted at an arbitrary node with lower height in the n-dimensional augmented cube AQn . We propose a constructive algorithm with the time complexity O(N log N) to construct 2n − 1 EISTs rooted at any node with the same height n in AQn (n ≥ 2), where N = 2n is the number of nodes in AQn . Acknowledgments This work is supported by National Natural Science Foundation of China (No. 61572337, No. 61872257, No. 61602333, and No. 61972272), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18 kJA520009), China Postdoctoral Science Foundation Funded Project (No. 2015M581858), the Jiangsu Planned Projects for Postdoctoral Research Funds, China (No. 1501089B), Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks Foundation, China (No. WSNLBKF201701), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China. References [1] F. Bao, Y. Funyu, Y. Hamada, Y. Igarashi, Reliable broadcasting and secure distributing in channel networks, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E81-A (1998) 796–806. [2] F. Bao, Y. Igarashi, S.R. Öhring, Reliable broadcasting in product networks, Discrete Appl. Math. 83 (1–3) (1998) 3–20. [3] H.-C. Chan, J.-M. Chang, Y.-L. Wang, S.-J. Horng, Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes, Appl. Math. Comput. 207 (2009) 333–339. [4] Y.S. Chen, C.Y. Chiang, C.Y. Chen, Multi-node broadcasting in all-ported 3-D wormhole-routed torus using an aggregation-then-distribution strategy, J. Syst. Architect. 50 (9) (2004) 575–589. [5] B. Cheng, J. Fan, X. Jia, J. Jia, Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes, J. Supercomput. 65 (3) (2013) 1279–1301. [6] B. Cheng, J. Fan, X. Jia, J. Wang, Dimension-adjacent trees and parallel construction of independent spanning trees on crossed cubes, J. Parallel Distrib. Comput. 73 (5) (2013) 641–652. [7] B. Cheng, J. Fan, X. Jia, S. Zhang, Independent spanning trees in crossed cubes, Inform. Sci. 233 (1) (2013) 276–289. [8] B. Cheng, J. Fan, X. Jia, S. Zhang, B. Chen, Constructive algorithm of independent spanning trees on Möbius cubes, Comput. J. 56 (11) (2013) 1347–1362. [9] B. Cheng, J. Fan, Q. Lyu, J. Zhou, Z. Liu, Constructing independent spanning trees with height n on the n-dimensional crossed cube, Futur. Gener. Comput. Syst. 87 (2018) 404–415. [10] D. Cheng, R.-X. Hao, Y.-Q. Feng, Conditional edge-fault pancyclicity of augmented cubes, Theoret. Comput. Sci. 510 (2013) 94–101. [11] J. Cheriyan, S.N. Maheshwari, Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs, J. Algorithms 9 (4) (1988) 507–537. [12] S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84. [13] S. Curran, O. Lee, X. Yu, Finding four independent trees, SIAM J. Comput. 35 (5) (2006) 1023–1058. [14] Q. Dong, X. Wang, How many triangles and quadrilaterals are there in an n-dimensional augmented cube, Theoret. Comput. Sci. 771 (2019) 93–98. [15] T. Elhourani, A. Gopalan, S. Ramasubramanian, IP Fast rerouting for multi-link failures, IEEE/ACM Trans. Netw. 24 (5) (2016) 3014–3025. [16] J.-S. Fu, Edge-fault-tolerant vertex-pancyclicity of augmented cubes, Inf. Process. Lett. 110 (2010) 439–443. [17] Z.-Y. Ge, S.L. Hakimi, Disjoint rooted spanning trees with small depths in deBruijn and Kautz graphs, SIAM J. Comput. 26 (1) (1997) 79–92. [18] A. Gopalan, S. Ramasubramanian, On constructing three edge independent spanning trees, Manscript (2011) (http://citeseerx.ist.psu.edu/viewdoc/ summary?doi=10.1.1.406.7119&rank=1). [19] A. Gopalan, S. Ramasubramanian, IP Fast rerouting and disjoint multipath routing with three edge-independent spanning trees, IEEE/ACM Trans. Netw. 24 (3) (2016) 1336–1349. [20] A. Hoyer, R. Thomas, Four edge-independent spanning tree, SIAM J. Discrete Math. 32 (1) (2018) 233–248. [21] S.-Y. Hsieh, Y.-R. Cian, Conditional edge-fault hamiltonicity of augmented cubes, Inform. Sci. 180 (2010) 2596–2617. [22] S.-Y. Hsieh, C.-J. Tu, Constructing edge-disjoint spanning trees in locally twisted cubes, Theoret. Comput. Sci. 410 (8–10) (2009) 926–932. [23] S.-Y. Hsieh, C.-Y. Wu, Edge-fault-tolerant hamiltonicity of locally twisted cubes under conditional edge faults, J. Comb. Optim. 19 (1) (2010) 16–30. [24] A. Huck, Independent trees in graphs, Graphs Combin. 10 (1) (1994) 29–45. [25] A. Huck, Independent trees in planar graphs, Graphs Combin. 15 (1) (1999) 29–77. [26] A. Itai, M. Rodeh, The multi-tree approach to reliability in distributed networks, Inform. Comput. 79 (1988) 43–59. [27] Y. Iwasaki, Y. Kajiwara, K. Obokata, Y. Igarashi, Independent spanning trees of chordal rings, Inform. Process. Lett. 69 (3) (1999) 155–160. [28] J.-S. Kim, H.-O. Lee, E. Cheng, L. László, Independent spanning trees on even networks, Inform. Sci. 181 (13) (2011) 2892–2905. [29] J.-S. Kim, H.-O. Lee, E. Cheng, L. László, Optimal independent spanning trees on odd graphs, J. Supercomput. 56 (2) (2011) 212–225. [30] Y.-J. Liu, W.Y. Chou, J.K. Lan, C. Chen, Constructing independent spanning trees for locally twisted cubes, Theoret. Comput. Sci. 412 (22) (2011) 2237–2252. [31] M. Ma, G. Liu, J.-M. Xu, The super connectivity of augmented cubes, Inf. Process. Lett. 106 (2008) 59–63. [32] S.A. Mane, S.A. Kandekar, B.N. Waphare, Constructing spanning trees in augmented cubes, J. Parallel Distrib. Comput. 122 (2018) 188–194. [33] K. Miura, D. Takahashi, S. Nakano, T. Nishizeki, A linear-time algorithm to find four independent spanning trees in four-connected planar graphs, Int. J. Found. Comput. S. 10 (2) (1999) 195–210. [34] S. Nagai, S. Nakano, A linear-time algorithm to find independent spanning trees in maximal planar graphs, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E84-A (5) (2001) 1102–1109. [35] K. Obokata, Y. Iwasaki, F. Bao, Y. Igarashi, Independent spanning trees of product graphs and their construction, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E79-A (11) (1996) 1894–1903. [36] S.-M. Tang, Y.-L. Wang, Y.-H. Leu, Optimal independent spanning trees on hypercubes, J. Inf. Sci. Eng. 20 (2004) 143–155.

Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.

16

B. Cheng, J. Fan, C.-K. Lin et al. / Discrete Applied Mathematics xxx (xxxx) xxx

[37] S.-M. Tang, J.-S. Yang, Y.-L. Wang, J.-M. Chang, Independent spanning trees on multidimensional torus networks, IEEE Trans. Comput. 59 (1) (2010) 93–102. [38] Y.-C. Tseng, S.-Y. Wang, C.-W. Ho, Efficient broadcasting in wormhole-routed multicomputers: A network-partitioning approach, IEEE Trans. Parallel Distrib. Syst. 10 (1) (1999) 44–61. [39] X. Wang, J. Fan, X. Jia, C.-K. Lin, An efficient algorithm to construct disjoint path covers of dcell networks, Theoret. Comput. Sci. 609 (2016) 197–210. [40] X. Wang, J. Fan, C.-K. Lin, X. Jia, Vertex-disjoint paths in dcell networks, J. Parallel Distrib. Comput. 96 (2016) 38–44. [41] Y. Wang, J. Fan, G. Zhou, X. Jia, Independent spanning trees on twisted cubes, J. Parallel Distrib. Comput. 72 (1) (2012) 58–69. [42] Y. Wang, H. Shen, J. Fan, Edge-independent spanning trees in augmented cubes, Theoret. Comput. Sci. 670 (2017) 23–32. [43] M.-C. Yang, Constructing edge-disjoint spanning trees in twisted cubes, Inform. Sci. 180 (20) (2010) 4075–4083. [44] J.-S. Yang, J.-M. Chang, Independent spanning trees on folded hyper-stars, Networks 56 (4) (2010) 44–53. [45] J.-S. Yang, J.-M. Chang, H.-C. Chan, Independent spanning trees on folded hypercubes, in: Proceedings of 10th International Symposium on Pervasive Systems, Algorithms, and Networks, I-SPAN 2009, Kaohsiung, Taiwan, 2009, pp. 73–83. [46] J.-S. Yang, J.-M. Chang, S.-M. Tang, Y.-L. Wang, Reducing the height of independent spanning trees in chordal rings, IEEE Trans. Parallel Distrib. Syst. 18 (5) (2007) 644–657. [47] J.-S. Yang, J.-M. Chang, S.-M. Tang, Y.-L. Wang, On the independent spanning trees of recursive circulant graphs G(cdm , d) with d > 2, Theoret. Comput. Sci. 410 (21–23) (2009) 2001–2010. [48] A. Zehavi, A. Itai, Three tree-paths, J. Graph Theory 13 (2) (1989) 175–188.

Please cite this article as: B. Cheng, J. Fan, C.-K. Lin et al., An improved algorithm to construct edge-independent spanning trees in augmented cubes, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.09.021.