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Constructing two completely independent spanning trees in hypercube-variant networks
JID:TCS AID:10914 /FLA
Doctopic: Algorithms, automata, complexity and games
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Theoretical Computer Science ••• (••••) •••–•••
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Theoretical Computer Science www.elsevier.com/locate/tcs
Constructing two completely independent spanning trees in hypercube-variant networks ✩ Kung-Jui Pai a,∗ , Jou-Ming Chang b a b
Department of Industrial Engineering and Management, Ming Chi University of Technology, New Taipei City, Taiwan, ROC Institute of Information and Decision Sciences, National Taipei University of Business, Taipei, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 19 May 2016 Received in revised form 31 July 2016 Accepted 29 August 2016 Available online xxxx Communicated by S.-Y. Hsieh Keywords: Completely independent spanning trees Hypercube-variant networks CIST-partition Diameter
a b s t r a c t Given a graph G, a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x, y ∈ V (G ), the paths joining x and y on these trees have neither vertex nor edge in common, except x and y. Hasunuma [9,10] first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, Péterfalvi [16] disproved this conjecture by constructing a k-connected graph, for each k 2, which does not have two CISTs. In this paper, we provide a unified approach for constructing two CISTs in several hypercube-variant networks, including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Möbius cubes. In particular, for an n-dimensional hypercube-variant network, the diameters of the constructed CISTs are 2n − 1. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The sets of vertices and edges of a graph G are denoted by V (G ) and E (G ), respectively. For two vertices x, y ∈ V (G ), an (x, y )-path is a path connecting x and y. Two (x, y )-paths are openly disjoint if they have no vertex and edge in common except for x and y. A set of k( 2) spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x, y ∈ V (G ), the (x, y )-paths in the k spanning trees are pairwise openly disjoint. Also, we say that two spanning trees T 1 and T 2 are edge-disjoint if they have no common edge. The study of CISTs was due to the early work of Hasunuma [9,10]. He showed that there are k CISTs in the underlying graph of any k-connected line digraph [9], there are two CISTs in any 4-connected maximal planar graph [10], and there are two CISTs in the Cartesian product of any 2-connected graphs [12]. Moreover, Hasunuma [10] showed that determining whether a graph G admits k CISTs is NP-complete, even for k = 2. In addition, a conjecture arisen from [10] stated that there are k CISTs in any 2k-connected graph. Later on, Péterfalvi [16] disproved this conjecture by showing that for any k 2, there exists a k-connected graph which does not possess two CISTs. For more counterexamples of graphs related to interconnection networks that fail to Hasunuma’s conjecture, see also [15]. Moreover, degree-based conditions that are sufficient for graphs admitting CISTs were recently investigated in [1,2,8,11]. Hypercube is a popular and more attractive interconnection network. To achieve some improvements of structure properties in contrast to hypercubes, many hypercube-variant networks have been proposed, e.g., locally twisted cubes [21],
✩ This research was partially supported by MOST grants 104-2221-E-131-004 (Kung-Jui Pai) and 104-2221-E-141-002-MY3 (Jou-Ming Chang), from the Ministry of Science and Technology, Taiwan. Corresponding author. E-mail address: [email protected] (K.-J. Pai).
*
http://dx.doi.org/10.1016/j.tcs.2016.08.024 0304-3975/© 2016 Elsevier B.V. All rights reserved.
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crossed cubes [7], parity cubes [17], and Möbius cubes [5]. Since the diameter is defined to be the greatest distance between any pair of vertices in a graph, it indicates that the worst case of the communication delay in the network. A main merits of the above-mentioned hypercube-variant networks is that their diameters are about half of those of the corresponding hypercubes. More previous results on the variations of hypercubes can also refer to [6,14]. In particular, a result in [15] showed that Hasunuma’s conjecture does not hold for n-dimensional hypercube and its variants when n is even and n = 2r for some certain positive integer r (e.g., Q 10 does not contain 5 CISTs). By contrast, Cheng et al. [3,4] recently proposed algorithms to construct two CISTs in locally twisted cubes and crossed cubes. In this paper, we provide a unified approach for constructing two CISTs in the above-mentioned hypercube-variant networks. For attaining to a theoretical lower bound, a major challenge on the issue of constructing spanning trees in a network is to pursue the research goal of reducing the height or diameter of the trees (e.g., see [13,19,20]). Hence, we also pay our attention to the efficiency of diameters of the constructed CISTs. To show the correctness of our construction, it relies on the following characterizations. Theorem 1. (See [9].) T 1 , T 2 , . . . , T k are CISTs in a graph G if and only if they are edge-disjoint spanning tree and for any v ∈ V (G ), there is at most one T i such that v is not a leaf (i.e., v is an internal vertex of T i ). Theorem 2. (See [1].) A graph G admits k CISTs if and only if there is a partition of V (G ) into V 1 , V 2 , . . . , V k , which is called the CIST-partition, such that (i) for i ∈ {1, 2, . . . , k}, the subgraph of G induced by V i , denoted by G [ V i ], is connected; (ii) for distinct i , j ∈ {1, 2, . . . , k}, the bipartite graph with bipartition V i ∪ V j and edge set {(x, y ) ∈ E (G ) : x ∈ V 1 , y ∈ V 2 }, denoted by B ( V 1 , V 2 , G ), has no tree component. Note that the later condition in the above theorem is equivalent to that every connected component H of B ( V 1 , V 2 , G ) satisfies | E ( H )| | V ( H )|. 2. Two CISTs in 4-dimensional variant cubes In this section, we first construct two CISTs in a 4-dimensional hypercube and its variants. For clarity, the result of each variant will be presented separately in the subsequent subsections. For n 5, since the construction of two CISTs in the n-dimensional hypercube and its variants are obtained by a recursive fashion, the results presented in this section can be viewed as the induction bases. A general rule for constructing two CISTs in a large-scale hypercube and its variants is given in the next section. In what follows, we shall find it convenient to adopt the following notation. As usual, vertices in a hypercube and its variants are encoded by using binary strings. A binary string x of length n is denoted by xn−1 xn−2 · · · x0 , where xi ∈ {0, 1} for 0 i n − 1. Let ⊕ denote the modulo 2 addition. For two binary strings x and y of length n, define x + y = zn−1 zn−2 · · · z0 , where zi = xi ⊕ y i for 0 i n − 1. Let e i denote the n-bit binary string with ith bit being 1 and all other bits being 0. The complement of a bit xi is written as x¯ i . Clearly, x + e i = xn−1 xn−2 · · · xi +1 x¯ i xi −1 · · · x0 . Furthermore, for any integer p ∈ {0, 1}, we define p · x = xn −1 xn −2 · · · x0 where xi = p · xi (mod 2). Also, we use the notation G [ S ] to denote the subgraph of G induced by a subset S ⊂ V (G ), and G x the labeled graph obtained from G by prefixing the binary string of every vertex with x. 2.1. Two CISTs in Q 4 The n-dimensional hypercube, denoted by Q n , is the graph consisting of 2n vertices labeled by binary strings of length n, and two vertices in Q n are joined by an edge if and only if they differ in exactly one position. Also, if two adjacent vertices x, y ∈ V ( Q n ) have a different bit at position i, then x and y are said to be the i-neighbors to each other, and for notational convenience we write as N i (x) = y or N i ( y ) = x. Fig. 1(a) depicts hypercube Q 4 . Recall that we already mentioned that there exist two CISTs in the Cartesian product of any 2-connected graphs [12]. Since Q 4 is isomorphic to the Cartesian product of two 4-cycles, it admits two CISTs. In particular, for Q 4 , the diameter of each CIST constructed from [12] is 8. In the following, we provide another construction scheme of CISTs for Q 4 . The results we obtained are different from the ones in [12] and each CIST has diameter 7. According to Theorem 2, we partition V ( Q 4 ) into two sets V 1 = {0, 2, 3, 7, 8, 9, 11, 12} and V 2 = {1, 4, 5, 6, 10, 13, 14, 15}. Due to the rather small size of V 1 and V 2 , it is easy to verify from Fig. 1(b) that both Q 4 [ V 1 ] (i.e., the graph induced by black vertices) and Q 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 1(c) shows that B ( V 1 , V 2 , Q 4 ) contains two components. Since each component consists of 8 vertices and 8 edges, this shows that B ( V 1 , V 2 , Q 4 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition, and thus the two CISTs of Q 4 are shown in Fig. 1(d). Since | E ( Q 4 )| = 32 and there are 15 edges in each spanning tree, we note that the two edges (9, 11) and (13, 15) are absent in the two CISTs.
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Doctopic: Algorithms, automata, complexity and games
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Fig. 1. (a) Hypercube Q 4 ; (b) two connected graphs Q 4 [ V 1 ] and Q 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , Q 4 ); (d) two CISTs of Q 4 .
2.2. Two CISTs in LTQ 4 The n-dimensional locally twisted cube LTQ n is defined recursively as follows (see [21]): (1) LTQ 1 is the complete graph on two vertices labeled by 0 and 1. LTQ 2 is a graph consisting of four vertices with labels 00, 01, 10, 11 together with four edges (00, 01), (00, 10), (01, 11), and (10, 11). (2) For n 3, LTQ n is composed of two subcubes LTQ n0−1 and LTQ n1−1 such that each vertex x = 0xn−2 xn−3 · · · x0 ∈ V (LTQ n0−1 ) is connected with the vertex 1(xn−2 ⊕ x0 )xn−3 · · · x0 ∈ V (LTQ n1−1 ) by an edge.
In the above definition, we also denote N n−1 (x) = y or N n−1 ( y ) = x to mean that x and y are the (n − 1)-neighbors to each other. Fig. 2(a) depicts LTQ 4 . We partition V (LTQ 4 ) into two sets V 1 = {0, 3, 4, 5, 7, 8, 11, 12} and V 2 = {1, 2, 6, 9, 10, 13, 14, 15}. Fig. 2(b) shows that both LTQ 4 [ V 1 ] (i.e., the graph induced by black vertices) and LTQ 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 2(c) shows that B ( V 1 , V 2 , LTQ 4 ) contains two components. Since each component has a 4-cycle, this shows that B ( V 1 , V 2 , LTQ 4 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition, and thus LTQ 4 has two CISTs, as shown in Fig. 2(d). Note that the two edges (0, 8) and (2, 10) are absent in the two CISTs, and the diameter of each CIST is 7. 2.3. Two CISTs in CQ 4 Two binary strings x = x1 x0 and y = y 1 y 0 are pair-related, denoted x ∼ y, if and only if (x, y ) ∈ {(00, 00), (10, 10), (01, 11), (11, 01)}. The n-dimensional crossed cube CQ n is the labeled graph with the following recursive fashion (see [7]): (1) CQ 1 is the complete graph on two vertices with labels 0 and 1. (2) For n 2, CQ n is composed of two subcubes CQ n0−1 and CQ n1−1 such that two vertices x = 0xn−2 · · · x1 x0 ∈ V (CQ n0−1 )
and y = 1 yn−2 · · · y 1 y 0 ∈ V (CQ n1−1 ) are joined by an edge if and only if (i) xn−2 = yn−2 if n is even, and (ii) x2i +1 x2i ∼ y 2i +1 y 2i for 0 i < (n − 1)/2 , where x and y are called the (n − 1)-neighbors to each other, and denote as N n−1 (x) = y or N n−1 ( y ) = x.
Fig. 3(a) depicts CQ 4 . We partition V (CQ 4 ) into two sets V 1 = {0, 1, 4, 5, 7, 11, 12, 14} and V 2 = {2, 3, 6, 8, 9, 10, 13, 15}. Fig. 3(b) shows that both CQ 4 [ V 1 ] (i.e., the graph induced by black vertices) and CQ 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 3(c) shows that B ( V 1 , V 2 , CQ 4 ) has only one component that contains a 6-cycle (6, 7, 13, 11, 10, 14, 6), and thus has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 3(d) shows
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Fig. 2. (a) Locally twisted cube LTQ 4 ; (b) two connected graphs LTQ 4 [ V 1 ] and LTQ 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , LTQ 4 ); (d) two CISTs of LTQ 4 .
Fig. 3. (a) Crossed cube CQ 4 ; (b) two connected graphs CQ 4 [ V 1 ] and CQ 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , CQ 4 ); (d) two CISTs of CQ 4 .
the resulting CISTs of CQ 4 , where each CIST has diameter 7. Note that the two edges (5, 7) and (8, 10) are absent in the two CISTs. So far, the construction of two CISTs for CQ 4 seems to be very well. However, if we adopt the CISTs shown in Fig. 3(d) as the induction base, it will eventually obtain CISTs with larger diameter for high-dimensional crossed cubes. A precise analysis and an illustration will be given in Section 3. In Fig. 4, we modify the second CIST by replacing the edge (2, 3) with the edge (8, 10) to play a part of the induction base, and consequently, the diameter becomes 8.
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Doctopic: Algorithms, automata, complexity and games
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Fig. 4. An alternative construction of CISTs for CQ 4 , where the dashed line indicates the change.
Fig. 5. (a) Parity cube PQ 4 ; (b) two connected graphs PQ 4 [ V 1 ] and PQ 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , PQ 4 ); (d) two CISTs of PQ 4 .
2.4. Two CISTs in PQ 4 The n-dimensional parity cube, denoted by PQ n , is recursively defined as follows (see [17] and an extension [18]): (1) For n 3, PQ n is isomorphic to LTQ n . (2) For n 4, PQ n is composed of two subcubes PQ n0−1 and PQ n1−1 such that two vertices x = xn−1 xn−2 · · · x0 ∈ V (PQ n0−1 ) and y = yn−1 yn−2 · · · y 0 ∈ V (PQ n1−1 ) are joined by an edge if and only if 3
n− 2
y = x + e n −1 +
k =0
x2k · e 2k+1 +
n −2
xk · e 1 ,
k =2
where x and y are called the (n − 1)-neighbors to each other, and denote as N n−1 (x) = y or N n−1 ( y ) = x. Fig. 5(a) depicts PQ 4 . We partition V (PQ 4 ) into two sets V 1 = {0, 4, 7, 9, 11, 12, 14, 15} and V 2 = {1, 2, 3, 5, 6, 8, 10, 13}. Fig. 5(b) shows that both PQ 4 [ V 1 ] (i.e., the graph induced by black vertices) and PQ 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 5(c) shows that B ( V 1 , V 2 , PQ 4 ) has only one non-tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 5(d) shows the resulting CISTs of PQ 4 , where each CIST has diameter 7. Note that the two edges (0, 8) and (1, 11) are absent in the two CISTs. It is similar to the situation of CQ 4 that the CISTs shown in Fig. 5(d) are not suitable for serving as the induction base for PQ n when n 5. In Fig. 6, we offer another alternative to play a part of the construction. Note that the alternative is
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Fig. 6. An alternative construction of CISTs for PQ 4 .
Fig. 7. (a) Möbius cube MQ 4,0 ; (b) two connected graphs MQ 4,0 [ V 1 ] and MQ 4,0 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , MQ 4,0 ); (d) two CISTs of MQ 4,0 .
completely different from the original one. Due to the space saving, we omit the details and only show the result of CISTs. It is easy to check that each CIST has diameter 8, and the two edges (4, 14) and (13, 15) of PQ 4 are absent in the CISTs. 2.5. Two CISTs in MQ 4 There are two types of n-dimensional Möbius cube (denoted by MQ n ). The 0-type Möbius cube (denoted by MQ n,0 ) and the 1-type Möbius cube (denoted by MQ n,1 ) are defined recursively as follows (see [5]): (1) Both MQ 1,0 and MQ 1,1 are the complete graph on two vertices labeled by 0 and 1. (2) For n 2, both MQ n,0 and MQ n,1 are composed of MQ n0−1,0 and MQ n1−1,1 such that two vertices x = xn−1 xn−2 · · · x0 ∈ V (MQ n0−1,0 ) and y = yn−1 yn−2 · · · y 0 ∈ V (MQ n1−1,1 ) are joined by an edge in MQ n,0 (respectively, in MQ n,1 ) if and only if (i) xn−1 = y¯ n−1 = 0, and (ii) xi = y i (respectively, xi = y¯ i ) for all i = 0, 1, . . . , n − 2, where x and y are called the (n − 1)-neighbors to each other, and denote as Nn−1 (x) = y or Nn−1 ( y ) = x.
We first consider MQ 4,0 as shown in Fig. 7(a). Then, we partition V (MQ 4,0 ) into two sets V 1 = {0, 3, 4, 6, 7, 8, 11, 14} and V 2 = {1, 2, 5, 9, 10, 12, 13, 15}. Fig. 7(b) shows that both MQ 4,0 [ V 1 ] (i.e., the graph induced by black vertices) and MQ 4,0 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 7(c) shows that B ( V 1 , V 2 , MQ 4,0 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 7(d) shows the resulting CISTs of MQ 4,0 , where each CIST has diameter 7. Note that the two edges (8, 15) and (9, 11) are absent in the two CISTs. Next, we consider MQ 4,1 as shown in Fig. 8(a). We partition V (MQ 4,1 ) into two sets V 1 = {0, 1, 4, 6, 8, 9, 11, 12} and V 2 = {2, 3, 5, 7, 10, 13, 14, 15}. Fig. 8(b) shows that both MQ 4,1 [ V 1 ] (i.e., the graph induced by black vertices) and MQ 4,1 [ V 2 ]
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Doctopic: Algorithms, automata, complexity and games
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Fig. 8. (a) Möbius cube MQ 4,1 ; (b) two connected graphs MQ 4,1 [ V 1 ] and MQ 4,1 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , MQ 4,1 ); (d) two CISTs of MQ 4,1 .
(i.e., the graph induced by gray vertices) are connected. Fig. 8(c) shows that B ( V 1 , V 2 , MQ 4,1 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 8(d) shows the resulting CISTs of MQ 4,1 , where each CIST has diameter 7. Note that the two edges (7, 8) and (12, 15) are absent in the two CISTs. From above, all 4-dimensional hypercube-variant networks we discussed have been shown to admit two CISTs. We close this section by giving the following property. Theorem 3. For 4-dimensional hypercube-variant networks, such as Q 4 , LTQ 4 , CQ 4 , PQ 4 and MQ 4 , there exist two CISTs with diameter 7. 3. A unified approach for constructing CISTs In this section, we give a unified approach to construct two CISTs in n-dimensional hypercubes and their variants for n 5. We describe the general rule as follows. Theorem 4. Let G n be the n-dimensional variant hypercube for n 4 and suppose that T 1 and T 2 are two CISTs of G n . For i ∈ {1, 2}, let Tˆ i be a spanning tree of G n+1 constructed from T i0 and T i1 by adding an edge (u i , v i ) ∈ E (G n+1 ) to connect two internal vertices u i ∈ V ( T i0 ) and v i ∈ V ( T i1 ). Then, Tˆ 1 and Tˆ 2 are two CISTs of G n+1 .
j
j
Proof. Recall that by Theorem 1, T 1 and T 2 are edge-disjoint spanning tree in G n , and thus for each j ∈ {0, 1}, T 1 and T 2
are edge-disjoint in Clearly, ∪ and ∪ E ( T 21 ) are disjoint. For i ∈ {1, 2}, let Tˆ i be a spanning tree of G n+1 described as above. By Theorem 1, if a vertex w ∈ V (G n ) is an internal vertex of T 1 (respectively, T 2 ), then it must be a leaf in T 2 (respectively, T 1 ). Since u i and v i are internal vertices of T i0 and T i1 for i ∈ {1, 2}, respectively, it follows that (u 1 , v 1 ) and (u 2 , v 2 ) are distinct. Moreover, we have (u 1 , v 1 ) ∈ / E ( T 20 ) ∪ E ( T 21 ) and (u 2 , v 2 ) ∈ / E ( T 10 ) ∪ E ( T 11 ). Since E ( Tˆ 1 ) = E ( T 10 ) ∪ E ( T 11 ) ∪ {(u 1 , v 1 )} and E ( Tˆ 2 ) = E ( T 20 ) ∪ E ( T 21 ) ∪ {(u 2 , v 2 )}, this shows that Tˆ 1 and Tˆ 2 are two edge-disjoint j Gn .
E ( T 10 )
E ( T 11 )
E ( T 20 )
spanning tree in G n+1 . Next, we show that every vertex of G n+1 must be a leaf in Tˆ 1 or Tˆ 2 . For each vertex w ∈ V (G n ), the vertex with prefixing a bit j ∈ {0, 1} in the label of w is denoted by w j . By Theorem 1, we know that there is at least one of T 1 and T 2 such that j it contains w as a leaf. Clearly, if w is a leaf in T i for i ∈ {1, 2}, then so is w j in T i , and this further implies that w j is still a leaf in Tˆ i because Tˆ i is constructed from T i0 and T i1 by adding an edge between two internal vertices of the trees. Thus, this proves the theorem. 2
For the facilitate of discussion, for i ∈ {1, 2}, such vertices u i ∈ V ( T i0 ) and v i ∈ V ( T i1 ) in the above theorem are called
the port vertices of the trees T i0 and T i1 , respectively. Moreover, the edge (u i , v i ) ∈ E (G n+1 ) is called the bridge of Tˆ i for the
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Fig. 9. An illustration of port vertices for pairs {0, 2k } where 4 k n.
construction. Obviously, choosing an appropriate pair {u i , v i } as port vertices in the construction is important because it will determine the diameter of the resulting tree. For example, we consider the construction of two CISTs in Q 5 . Let T 1 and T 2 be two CISTs of the 4-dimensional hypercube constructed in the previous section (see Fig. 1(d)), where T 1 and T 2 take the black vertices as internal vertices and leaves, respectively. Clearly, N 4 (12) = 12 + 24 = 28. If we choose the pair {12, 28} as the port vertices for T 10 and T 11 ,
then both the trees T 10 , T 11 , together with the bridge (12, 28) form a spanning tree Tˆ 1 of Q 5 with diameter 13. Indeed, if we choose vertices near the center of the trees as their port vertices, then we can get a better result of the diameter of CISTs for the construction. Note that the center of a graph is defined to be the set of vertices which minimize the maximal distance from other vertices in the graph. In particular, the cardinality of the center in a tree is at most two. In the case of Q 4 , we would like to choose the pair {0, 16(= 0 + 24 )} as the port vertices for T 10 and T 11 . Similarly, we choose the pair {4, 20(= 4 + 24 )} as the port vertices for T 20 and T 21 . As a result, by Theorem 4, the two CISTs constructed for Q 5 have diameter 9. In general, we construct two CISTs in Q n+1 for n 4 as follows. Suppose that T 1 and T 2 are two CISTs of Q n and let Tˆ 1 and Tˆ 2 be two CISTs of Q n+1 constructed from the rule of Theorem 4. Clearly, for i ∈ {1, 2}, if a vertex u is a center vertex of T i0 , then so is the vertex N n (u ) = u + 2n of T i1 . Thus, the choice of pairs {0, 2n } and {4, 4 + 2n } as port vertices can build two CISTs of Q n+1 by induction on n. See Fig. 9 for an illustration of port vertices in a recursive construction. Since we can always choose center vertices of T 1 and T 2 as port vertices, it follows that
D ( Tˆ i ) = 2 ·
D (T i ) 2
+ 1 for i ∈ {1, 2},
(1)
where D ( T ) denotes the diameter of the tree T . Therefore, by Theorems 3 and 4, we have the following corollary. Corollary 5. For n 4, Q n admits two CISTs with diameter 2n − 1. In what follows, we consider the construction of two CISTs in other high-dimensional hypercube-variant networks. For a 4-dimensional variant hypercube, such as LTQ 4 , CQ 4 , PQ 4 and MQ 4,i for i ∈ {0, 1}, we assume that T 1 and T 2 are two CISTs constructed in the previous section, where T 1 and T 2 take the black vertices as internal vertices and leaves, respectively. For constructing two CISTs in LTQ n+1 with n 4, we first observe the two CISTs of LTQ 4 in Fig. 2(d). It shows that vertices 4 and 5 are in the center of T 1 , and vertices 14 and 15 are in the center of T 2 . Recall that the n-neighbor of a vertex x in LTQ n+1 is defined to be N n (x) = x¯ n (xn−1 ⊕ x0 )xn−2 · · · x0 . Hence, for arbitrary n 4, we have N n (4) = 4 + 2n and N n (14) = 14 + 2n in LTQ n+1 . According to Theorem 4, we can choose the pairs {4, 4 + 2n } and {14, 14 + 2n } as port vertices for building two CISTs of LTQ n+1 by using induction on n. Thus, through an argument similar to the analysis of the diameter of CISTs in hypercubes, we obtain the following corollary. Corollary 6. For n 4, LTQ n admits two CISTs with diameter 2n − 1. For constructing two CISTs in CQ n+1 with n 4, we observe the two CISTs of CQ 4 in Fig. 3(d), and it shows that vertices 0 and 4 are in the center of T 1 , and vertices 3 and 9 are in the center of T 2 . From the definition of the n-neighbor in
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CQ n+1 , we have N n (0) = 2n . Thus, by Theorem 4, we can choose the pair {0, 2n } as port vertices for building the first CIST of CQ n+1 by using induction on n. To build the second CIST of CQ n+1 , unfortunately, if we choose vertex 3 (respectively, vertex 9) as port vertex of a CIST in CQ n0 , then the vertex N n (3) = (3 − 21 ) + 2n (respectively, N n (9) = (9 + 21 ) + 2n ) is a leaf in the corresponding tree of CQ n1 , and thus cannot serve as a port vertex. So, we consider an alternative as shown in Fig. 4, where T 1 is the same as before and T 2 contains only one center vertex 8. Since N n (8) = 8 + 2n , we can choose the pair {8, 8 + 2n } as port vertices for building the second CIST of CQ n+1 . According to Theorems 3, 4, and Eq. (1), we have the following corollary. Corollary 7. For n 4, CQ n admits two CISTs with diameter 2n − 1. For constructing two CISTs in PQ n+1 with n 4, we observe the two CISTs of PQ 4 in Fig. 5(d), and it shows that vertices 14 and 15 are in the center of T 1 , and vertices 2 and 3 are in the center of T 2 . From the definition of the n-neighbor in PQ n+1 , we have N n (14) = (14 − 23 ) + 2n , N n (15) = (15 − 23 − 21 ) + 2n , N n (2) = 2 + 2n , and N n (3) = (3 − 21 ) + 2n . In this case, we can choose the pair {2, 2 + 2n } as port vertices for building the second CIST. However, since both N 4 (14) and N 4 (15) are leaves in T 11 , none of them can serve as port vertex for building the first CIST of PQ 5 . Similar to the situation of crossed cubes, we consider an alternative as shown in Fig. 6. In this construction, it shows that T 1 contains only one center vertex 0 and T 2 contains only one center vertex 2. Since N n (0) = 2n and N n (2) = 2 + 2n , we can choose the pairs {0, 2n } and {2, 2 + 2n } as port vertices for building two CISTs of PQ n by using induction on n. Again, by Theorems 3, 4, and Eq. (1), we obtain the following corollary. Corollary 8. For n 4, PQ n admits two CISTs with diameter 2n − 1. Finally, we consider the construction of two CISTs in MQ n+1 for n 4. Let T 1 and T 2 be two CISTs of MQ 4,0 as shown in Fig. 7(d), where vertices 4 and 7 are in the center of T 1 , and vertices 5 and 13 are in the center of T 2 . Also, let T 1 and T 2 be two CISTs of MQ n,1 as shown in Fig. 8(d), where vertices 4 and 11 are in the center of T 1 , and vertices 2 and 13 are in the center of T 2 . Recall that the n-neighbor of a vertex x in MQ n+1,0 (respectively, in MQ n+1,1 ) is defined n n to be N n (x) = x¯ n xn−1 · · · x0 (respectively, N n (x) = x¯ n x¯ n−1 · · · x¯ 0 ). Hence, n nwe have Nn (4) = 4 + 2 and Nn (13) = 13 + 2 in MQ n+1,0 , and N n (4) = (15 − 4) + k=4 2k and N n (13) = (15 − 13) + k=4 2k in MQ n+1,1 . To construct two CISTs of MQ 5,0 , we choose the pairs {4, 20(= 4 + 24 )} and {13, 29(= 13 + 24 )} as port vertices, where 4 ∈ V ( T 10 ), 20 ∈ V ( T 11 ), 13 ∈ V ( T 20 ), and 29 ∈ V ( T 21 ) are all center vertices in the trees. To construct two CISTs of MQ 5,1 , we choose the pairs {4, 27(= 11 + 24 )} and {13, 18(= 2 + 24 )} as port vertices, where 4 ∈ V ( T 10 ), 27 ∈ V ( T 11 ), 13 ∈ V ( T 20 ), and 18 ∈ V ( T 21 ) are all center vertices in the trees. In general, we can choose the pairs {4, 4 + 2n } and {13, 13 + 2n } as port vertices for building two CISTs of MQ n+1,0 n n by using induction on n. By contrast, we can choose the pairs {4, 11 + k=4 2k } and {13, 2 + k=4 2k } as port vertices for building two CISTs of MQ n+1,1 by using induction on n. As a result, by Theorems 3, 4, and Eq. (1), we obtain the following corollary. Corollary 9. For n 4 and i ∈ {0, 1}, MQ n,i admits two CISTs with diameter 2n − 1. 4. Concluding remarks In this paper, we provide a unified approach for constructing two CISTs in hypercubes and their variants. In addition, we give the exact diameter of the constructed CISTs. Since research results related to CISTs are limited, the following three problems are worthwhile to be addressed in a future research. (1) Is it possible to reduce the diameter of CISTs for hypercube-variant networks mentioned above? (2) How to design algorithms to construct more (e.g. 3) CISTs in high-dimensional hypercube-variant networks? Especially, is there a unified approach? (3) Does there exist a condition of necessity and sufficiency for which Hasunuma’s conjecture is affirmative in a subclass of Q n (or other variants)? References [1] T. Araki, Dirac’s condition for completely independent spanning trees, J. Graph Theory 77 (2014) 171–179. [2] H.-Y. Chang, H.-L. Wang, J.-S. Yang, J.-M. Chang, A note on the degree condition of completely independent spanning trees, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 98-A (2015) 2191–2193. [3] B. Cheng, J. Fan, D. Wang, J. Yang, A reliable broadcasting algorithm in locally twisted cubes, in: Proc. 2nd Int. Conf. on Cyber Security and Cloud Computing, 2015, pp. 323–328. [4] B. Cheng, D. Wang, J. Fan, Constructing completely independent spanning trees in crossed cubes, personal communication. [5] P. Cull, S.L. Larson, The Möbius cubes, IEEE Trans. Comput. 44 (1995) 647–659. [6] K. Efe, A variation on the hypercube with lower diameter, IEEE Trans. Comput. 40 (1991) 1312–1316. [7] K. Efe, The crossed cube architecture for parallel computation, IEEE Trans. Parallel Distrib. Syst. 3 (1992) 513–524.
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[8] G. Fan, Y. Hong, Q. Liu, Ore’s condition for completely independent spanning trees, Discrete Appl. Math. 177 (2014) 95–100. [9] T. Hasunuma, Completely independent spanning trees in the underlying graph of a line digraph, Discrete Math. 234 (2001) 149–157. [10] T. Hasunuma, Completely independent spanning trees in maximal planar graphs, in: Proc. 28th Int. Workshop on Graph-Theoretic Concepts in Computer Science, WG 2002, in: Lecture Notes in Comput. Sci., vol. 2573, 2002, pp. 235–245. [11] T. Hasunuma, Minimum degree conditions and optimal graphs for completely independent spanning trees, in: Proc. 26th Int. Workshop on Combinatorial Algorithms, IWOCA 2016, in: Lecture Notes in Comput. Sci., vol. 9538, 2016, pp. 260–273. [12] T. Hasunuma, C. Morisaka, Completely independent spanning trees in torus networks, Networks 60 (2012) 59–69. [13] T. Hasunuma, H. Nagamochi, Independent spanning trees with small depths in iterated line digraphs, Discrete Appl. Math. 110 (2001) 189–211. [14] K.S. Hu, S.-S. Yeoh, C. Chen, L.-H. Hsu, Node-pancyclicity and edge-pancyclicity of hypercube variants, Inform. Process. Lett. 102 (2007) 1–7. [15] K.-J. Pai, J.-S. Yang, S.-C. Yao, S.-M. Tang, J.-M. Chang, Completely independent spanning trees on some interconnection networks, IEICE Trans. Inf. Syst. 97-D (2014) 2514–2517. [16] F. Péterfalvi, Two counterexamples on completely independent spanning trees, Discrete Math. 312 (2012) 808–810. [17] D. Wang, L. Zhao, The twisted-cube connected networks, J. Comput. Sci. Tech. 14 (1999) 181–187. [18] Y. Wang, J. Fan, X. Jia, H. Huang, An algorithm to construct independent spanning trees on parity cubes, Theoret. Comput. Sci. 465 (2012) 61–72. [19] 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 (2007) 644–657. [20] J.-S. Yang, S.-S. Luo, J.-M. Chang, Pruning longer branches of independent spanning trees on folded hyper-stars, Comput. J. 58 (2015) 2979–2981. [21] X. Yang, D.J. Evans, G.M. Megson, The locally twisted cubes, Int. J. Comput. Math. 82 (2005) 401–413.
Doctopic: Algorithms, automata, complexity and games
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Theoretical Computer Science ••• (••••) •••–•••
Contents lists available at ScienceDirect
Theoretical Computer Science www.elsevier.com/locate/tcs
Constructing two completely independent spanning trees in hypercube-variant networks ✩ Kung-Jui Pai a,∗ , Jou-Ming Chang b a b
Department of Industrial Engineering and Management, Ming Chi University of Technology, New Taipei City, Taiwan, ROC Institute of Information and Decision Sciences, National Taipei University of Business, Taipei, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 19 May 2016 Received in revised form 31 July 2016 Accepted 29 August 2016 Available online xxxx Communicated by S.-Y. Hsieh Keywords: Completely independent spanning trees Hypercube-variant networks CIST-partition Diameter
a b s t r a c t Given a graph G, a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x, y ∈ V (G ), the paths joining x and y on these trees have neither vertex nor edge in common, except x and y. Hasunuma [9,10] first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, Péterfalvi [16] disproved this conjecture by constructing a k-connected graph, for each k 2, which does not have two CISTs. In this paper, we provide a unified approach for constructing two CISTs in several hypercube-variant networks, including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Möbius cubes. In particular, for an n-dimensional hypercube-variant network, the diameters of the constructed CISTs are 2n − 1. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The sets of vertices and edges of a graph G are denoted by V (G ) and E (G ), respectively. For two vertices x, y ∈ V (G ), an (x, y )-path is a path connecting x and y. Two (x, y )-paths are openly disjoint if they have no vertex and edge in common except for x and y. A set of k( 2) spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x, y ∈ V (G ), the (x, y )-paths in the k spanning trees are pairwise openly disjoint. Also, we say that two spanning trees T 1 and T 2 are edge-disjoint if they have no common edge. The study of CISTs was due to the early work of Hasunuma [9,10]. He showed that there are k CISTs in the underlying graph of any k-connected line digraph [9], there are two CISTs in any 4-connected maximal planar graph [10], and there are two CISTs in the Cartesian product of any 2-connected graphs [12]. Moreover, Hasunuma [10] showed that determining whether a graph G admits k CISTs is NP-complete, even for k = 2. In addition, a conjecture arisen from [10] stated that there are k CISTs in any 2k-connected graph. Later on, Péterfalvi [16] disproved this conjecture by showing that for any k 2, there exists a k-connected graph which does not possess two CISTs. For more counterexamples of graphs related to interconnection networks that fail to Hasunuma’s conjecture, see also [15]. Moreover, degree-based conditions that are sufficient for graphs admitting CISTs were recently investigated in [1,2,8,11]. Hypercube is a popular and more attractive interconnection network. To achieve some improvements of structure properties in contrast to hypercubes, many hypercube-variant networks have been proposed, e.g., locally twisted cubes [21],
✩ This research was partially supported by MOST grants 104-2221-E-131-004 (Kung-Jui Pai) and 104-2221-E-141-002-MY3 (Jou-Ming Chang), from the Ministry of Science and Technology, Taiwan. Corresponding author. E-mail address: [email protected] (K.-J. Pai).
*
http://dx.doi.org/10.1016/j.tcs.2016.08.024 0304-3975/© 2016 Elsevier B.V. All rights reserved.
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crossed cubes [7], parity cubes [17], and Möbius cubes [5]. Since the diameter is defined to be the greatest distance between any pair of vertices in a graph, it indicates that the worst case of the communication delay in the network. A main merits of the above-mentioned hypercube-variant networks is that their diameters are about half of those of the corresponding hypercubes. More previous results on the variations of hypercubes can also refer to [6,14]. In particular, a result in [15] showed that Hasunuma’s conjecture does not hold for n-dimensional hypercube and its variants when n is even and n = 2r for some certain positive integer r (e.g., Q 10 does not contain 5 CISTs). By contrast, Cheng et al. [3,4] recently proposed algorithms to construct two CISTs in locally twisted cubes and crossed cubes. In this paper, we provide a unified approach for constructing two CISTs in the above-mentioned hypercube-variant networks. For attaining to a theoretical lower bound, a major challenge on the issue of constructing spanning trees in a network is to pursue the research goal of reducing the height or diameter of the trees (e.g., see [13,19,20]). Hence, we also pay our attention to the efficiency of diameters of the constructed CISTs. To show the correctness of our construction, it relies on the following characterizations. Theorem 1. (See [9].) T 1 , T 2 , . . . , T k are CISTs in a graph G if and only if they are edge-disjoint spanning tree and for any v ∈ V (G ), there is at most one T i such that v is not a leaf (i.e., v is an internal vertex of T i ). Theorem 2. (See [1].) A graph G admits k CISTs if and only if there is a partition of V (G ) into V 1 , V 2 , . . . , V k , which is called the CIST-partition, such that (i) for i ∈ {1, 2, . . . , k}, the subgraph of G induced by V i , denoted by G [ V i ], is connected; (ii) for distinct i , j ∈ {1, 2, . . . , k}, the bipartite graph with bipartition V i ∪ V j and edge set {(x, y ) ∈ E (G ) : x ∈ V 1 , y ∈ V 2 }, denoted by B ( V 1 , V 2 , G ), has no tree component. Note that the later condition in the above theorem is equivalent to that every connected component H of B ( V 1 , V 2 , G ) satisfies | E ( H )| | V ( H )|. 2. Two CISTs in 4-dimensional variant cubes In this section, we first construct two CISTs in a 4-dimensional hypercube and its variants. For clarity, the result of each variant will be presented separately in the subsequent subsections. For n 5, since the construction of two CISTs in the n-dimensional hypercube and its variants are obtained by a recursive fashion, the results presented in this section can be viewed as the induction bases. A general rule for constructing two CISTs in a large-scale hypercube and its variants is given in the next section. In what follows, we shall find it convenient to adopt the following notation. As usual, vertices in a hypercube and its variants are encoded by using binary strings. A binary string x of length n is denoted by xn−1 xn−2 · · · x0 , where xi ∈ {0, 1} for 0 i n − 1. Let ⊕ denote the modulo 2 addition. For two binary strings x and y of length n, define x + y = zn−1 zn−2 · · · z0 , where zi = xi ⊕ y i for 0 i n − 1. Let e i denote the n-bit binary string with ith bit being 1 and all other bits being 0. The complement of a bit xi is written as x¯ i . Clearly, x + e i = xn−1 xn−2 · · · xi +1 x¯ i xi −1 · · · x0 . Furthermore, for any integer p ∈ {0, 1}, we define p · x = xn −1 xn −2 · · · x0 where xi = p · xi (mod 2). Also, we use the notation G [ S ] to denote the subgraph of G induced by a subset S ⊂ V (G ), and G x the labeled graph obtained from G by prefixing the binary string of every vertex with x. 2.1. Two CISTs in Q 4 The n-dimensional hypercube, denoted by Q n , is the graph consisting of 2n vertices labeled by binary strings of length n, and two vertices in Q n are joined by an edge if and only if they differ in exactly one position. Also, if two adjacent vertices x, y ∈ V ( Q n ) have a different bit at position i, then x and y are said to be the i-neighbors to each other, and for notational convenience we write as N i (x) = y or N i ( y ) = x. Fig. 1(a) depicts hypercube Q 4 . Recall that we already mentioned that there exist two CISTs in the Cartesian product of any 2-connected graphs [12]. Since Q 4 is isomorphic to the Cartesian product of two 4-cycles, it admits two CISTs. In particular, for Q 4 , the diameter of each CIST constructed from [12] is 8. In the following, we provide another construction scheme of CISTs for Q 4 . The results we obtained are different from the ones in [12] and each CIST has diameter 7. According to Theorem 2, we partition V ( Q 4 ) into two sets V 1 = {0, 2, 3, 7, 8, 9, 11, 12} and V 2 = {1, 4, 5, 6, 10, 13, 14, 15}. Due to the rather small size of V 1 and V 2 , it is easy to verify from Fig. 1(b) that both Q 4 [ V 1 ] (i.e., the graph induced by black vertices) and Q 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 1(c) shows that B ( V 1 , V 2 , Q 4 ) contains two components. Since each component consists of 8 vertices and 8 edges, this shows that B ( V 1 , V 2 , Q 4 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition, and thus the two CISTs of Q 4 are shown in Fig. 1(d). Since | E ( Q 4 )| = 32 and there are 15 edges in each spanning tree, we note that the two edges (9, 11) and (13, 15) are absent in the two CISTs.
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Fig. 1. (a) Hypercube Q 4 ; (b) two connected graphs Q 4 [ V 1 ] and Q 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , Q 4 ); (d) two CISTs of Q 4 .
2.2. Two CISTs in LTQ 4 The n-dimensional locally twisted cube LTQ n is defined recursively as follows (see [21]): (1) LTQ 1 is the complete graph on two vertices labeled by 0 and 1. LTQ 2 is a graph consisting of four vertices with labels 00, 01, 10, 11 together with four edges (00, 01), (00, 10), (01, 11), and (10, 11). (2) For n 3, LTQ n is composed of two subcubes LTQ n0−1 and LTQ n1−1 such that each vertex x = 0xn−2 xn−3 · · · x0 ∈ V (LTQ n0−1 ) is connected with the vertex 1(xn−2 ⊕ x0 )xn−3 · · · x0 ∈ V (LTQ n1−1 ) by an edge.
In the above definition, we also denote N n−1 (x) = y or N n−1 ( y ) = x to mean that x and y are the (n − 1)-neighbors to each other. Fig. 2(a) depicts LTQ 4 . We partition V (LTQ 4 ) into two sets V 1 = {0, 3, 4, 5, 7, 8, 11, 12} and V 2 = {1, 2, 6, 9, 10, 13, 14, 15}. Fig. 2(b) shows that both LTQ 4 [ V 1 ] (i.e., the graph induced by black vertices) and LTQ 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 2(c) shows that B ( V 1 , V 2 , LTQ 4 ) contains two components. Since each component has a 4-cycle, this shows that B ( V 1 , V 2 , LTQ 4 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition, and thus LTQ 4 has two CISTs, as shown in Fig. 2(d). Note that the two edges (0, 8) and (2, 10) are absent in the two CISTs, and the diameter of each CIST is 7. 2.3. Two CISTs in CQ 4 Two binary strings x = x1 x0 and y = y 1 y 0 are pair-related, denoted x ∼ y, if and only if (x, y ) ∈ {(00, 00), (10, 10), (01, 11), (11, 01)}. The n-dimensional crossed cube CQ n is the labeled graph with the following recursive fashion (see [7]): (1) CQ 1 is the complete graph on two vertices with labels 0 and 1. (2) For n 2, CQ n is composed of two subcubes CQ n0−1 and CQ n1−1 such that two vertices x = 0xn−2 · · · x1 x0 ∈ V (CQ n0−1 )
and y = 1 yn−2 · · · y 1 y 0 ∈ V (CQ n1−1 ) are joined by an edge if and only if (i) xn−2 = yn−2 if n is even, and (ii) x2i +1 x2i ∼ y 2i +1 y 2i for 0 i < (n − 1)/2 , where x and y are called the (n − 1)-neighbors to each other, and denote as N n−1 (x) = y or N n−1 ( y ) = x.
Fig. 3(a) depicts CQ 4 . We partition V (CQ 4 ) into two sets V 1 = {0, 1, 4, 5, 7, 11, 12, 14} and V 2 = {2, 3, 6, 8, 9, 10, 13, 15}. Fig. 3(b) shows that both CQ 4 [ V 1 ] (i.e., the graph induced by black vertices) and CQ 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 3(c) shows that B ( V 1 , V 2 , CQ 4 ) has only one component that contains a 6-cycle (6, 7, 13, 11, 10, 14, 6), and thus has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 3(d) shows
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Fig. 2. (a) Locally twisted cube LTQ 4 ; (b) two connected graphs LTQ 4 [ V 1 ] and LTQ 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , LTQ 4 ); (d) two CISTs of LTQ 4 .
Fig. 3. (a) Crossed cube CQ 4 ; (b) two connected graphs CQ 4 [ V 1 ] and CQ 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , CQ 4 ); (d) two CISTs of CQ 4 .
the resulting CISTs of CQ 4 , where each CIST has diameter 7. Note that the two edges (5, 7) and (8, 10) are absent in the two CISTs. So far, the construction of two CISTs for CQ 4 seems to be very well. However, if we adopt the CISTs shown in Fig. 3(d) as the induction base, it will eventually obtain CISTs with larger diameter for high-dimensional crossed cubes. A precise analysis and an illustration will be given in Section 3. In Fig. 4, we modify the second CIST by replacing the edge (2, 3) with the edge (8, 10) to play a part of the induction base, and consequently, the diameter becomes 8.
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Fig. 4. An alternative construction of CISTs for CQ 4 , where the dashed line indicates the change.
Fig. 5. (a) Parity cube PQ 4 ; (b) two connected graphs PQ 4 [ V 1 ] and PQ 4 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , PQ 4 ); (d) two CISTs of PQ 4 .
2.4. Two CISTs in PQ 4 The n-dimensional parity cube, denoted by PQ n , is recursively defined as follows (see [17] and an extension [18]): (1) For n 3, PQ n is isomorphic to LTQ n . (2) For n 4, PQ n is composed of two subcubes PQ n0−1 and PQ n1−1 such that two vertices x = xn−1 xn−2 · · · x0 ∈ V (PQ n0−1 ) and y = yn−1 yn−2 · · · y 0 ∈ V (PQ n1−1 ) are joined by an edge if and only if 3
n− 2
y = x + e n −1 +
k =0
x2k · e 2k+1 +
n −2
xk · e 1 ,
k =2
where x and y are called the (n − 1)-neighbors to each other, and denote as N n−1 (x) = y or N n−1 ( y ) = x. Fig. 5(a) depicts PQ 4 . We partition V (PQ 4 ) into two sets V 1 = {0, 4, 7, 9, 11, 12, 14, 15} and V 2 = {1, 2, 3, 5, 6, 8, 10, 13}. Fig. 5(b) shows that both PQ 4 [ V 1 ] (i.e., the graph induced by black vertices) and PQ 4 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 5(c) shows that B ( V 1 , V 2 , PQ 4 ) has only one non-tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 5(d) shows the resulting CISTs of PQ 4 , where each CIST has diameter 7. Note that the two edges (0, 8) and (1, 11) are absent in the two CISTs. It is similar to the situation of CQ 4 that the CISTs shown in Fig. 5(d) are not suitable for serving as the induction base for PQ n when n 5. In Fig. 6, we offer another alternative to play a part of the construction. Note that the alternative is
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Fig. 6. An alternative construction of CISTs for PQ 4 .
Fig. 7. (a) Möbius cube MQ 4,0 ; (b) two connected graphs MQ 4,0 [ V 1 ] and MQ 4,0 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , MQ 4,0 ); (d) two CISTs of MQ 4,0 .
completely different from the original one. Due to the space saving, we omit the details and only show the result of CISTs. It is easy to check that each CIST has diameter 8, and the two edges (4, 14) and (13, 15) of PQ 4 are absent in the CISTs. 2.5. Two CISTs in MQ 4 There are two types of n-dimensional Möbius cube (denoted by MQ n ). The 0-type Möbius cube (denoted by MQ n,0 ) and the 1-type Möbius cube (denoted by MQ n,1 ) are defined recursively as follows (see [5]): (1) Both MQ 1,0 and MQ 1,1 are the complete graph on two vertices labeled by 0 and 1. (2) For n 2, both MQ n,0 and MQ n,1 are composed of MQ n0−1,0 and MQ n1−1,1 such that two vertices x = xn−1 xn−2 · · · x0 ∈ V (MQ n0−1,0 ) and y = yn−1 yn−2 · · · y 0 ∈ V (MQ n1−1,1 ) are joined by an edge in MQ n,0 (respectively, in MQ n,1 ) if and only if (i) xn−1 = y¯ n−1 = 0, and (ii) xi = y i (respectively, xi = y¯ i ) for all i = 0, 1, . . . , n − 2, where x and y are called the (n − 1)-neighbors to each other, and denote as Nn−1 (x) = y or Nn−1 ( y ) = x.
We first consider MQ 4,0 as shown in Fig. 7(a). Then, we partition V (MQ 4,0 ) into two sets V 1 = {0, 3, 4, 6, 7, 8, 11, 14} and V 2 = {1, 2, 5, 9, 10, 12, 13, 15}. Fig. 7(b) shows that both MQ 4,0 [ V 1 ] (i.e., the graph induced by black vertices) and MQ 4,0 [ V 2 ] (i.e., the graph induced by gray vertices) are connected. Fig. 7(c) shows that B ( V 1 , V 2 , MQ 4,0 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 7(d) shows the resulting CISTs of MQ 4,0 , where each CIST has diameter 7. Note that the two edges (8, 15) and (9, 11) are absent in the two CISTs. Next, we consider MQ 4,1 as shown in Fig. 8(a). We partition V (MQ 4,1 ) into two sets V 1 = {0, 1, 4, 6, 8, 9, 11, 12} and V 2 = {2, 3, 5, 7, 10, 13, 14, 15}. Fig. 8(b) shows that both MQ 4,1 [ V 1 ] (i.e., the graph induced by black vertices) and MQ 4,1 [ V 2 ]
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Fig. 8. (a) Möbius cube MQ 4,1 ; (b) two connected graphs MQ 4,1 [ V 1 ] and MQ 4,1 [ V 2 ]; (c) the bipartite graph B ( V 1 , V 2 , MQ 4,1 ); (d) two CISTs of MQ 4,1 .
(i.e., the graph induced by gray vertices) are connected. Fig. 8(c) shows that B ( V 1 , V 2 , MQ 4,1 ) has no tree component. By Theorem 2, V 1 and V 2 form a CIST-partition. Fig. 8(d) shows the resulting CISTs of MQ 4,1 , where each CIST has diameter 7. Note that the two edges (7, 8) and (12, 15) are absent in the two CISTs. From above, all 4-dimensional hypercube-variant networks we discussed have been shown to admit two CISTs. We close this section by giving the following property. Theorem 3. For 4-dimensional hypercube-variant networks, such as Q 4 , LTQ 4 , CQ 4 , PQ 4 and MQ 4 , there exist two CISTs with diameter 7. 3. A unified approach for constructing CISTs In this section, we give a unified approach to construct two CISTs in n-dimensional hypercubes and their variants for n 5. We describe the general rule as follows. Theorem 4. Let G n be the n-dimensional variant hypercube for n 4 and suppose that T 1 and T 2 are two CISTs of G n . For i ∈ {1, 2}, let Tˆ i be a spanning tree of G n+1 constructed from T i0 and T i1 by adding an edge (u i , v i ) ∈ E (G n+1 ) to connect two internal vertices u i ∈ V ( T i0 ) and v i ∈ V ( T i1 ). Then, Tˆ 1 and Tˆ 2 are two CISTs of G n+1 .
j
j
Proof. Recall that by Theorem 1, T 1 and T 2 are edge-disjoint spanning tree in G n , and thus for each j ∈ {0, 1}, T 1 and T 2
are edge-disjoint in Clearly, ∪ and ∪ E ( T 21 ) are disjoint. For i ∈ {1, 2}, let Tˆ i be a spanning tree of G n+1 described as above. By Theorem 1, if a vertex w ∈ V (G n ) is an internal vertex of T 1 (respectively, T 2 ), then it must be a leaf in T 2 (respectively, T 1 ). Since u i and v i are internal vertices of T i0 and T i1 for i ∈ {1, 2}, respectively, it follows that (u 1 , v 1 ) and (u 2 , v 2 ) are distinct. Moreover, we have (u 1 , v 1 ) ∈ / E ( T 20 ) ∪ E ( T 21 ) and (u 2 , v 2 ) ∈ / E ( T 10 ) ∪ E ( T 11 ). Since E ( Tˆ 1 ) = E ( T 10 ) ∪ E ( T 11 ) ∪ {(u 1 , v 1 )} and E ( Tˆ 2 ) = E ( T 20 ) ∪ E ( T 21 ) ∪ {(u 2 , v 2 )}, this shows that Tˆ 1 and Tˆ 2 are two edge-disjoint j Gn .
E ( T 10 )
E ( T 11 )
E ( T 20 )
spanning tree in G n+1 . Next, we show that every vertex of G n+1 must be a leaf in Tˆ 1 or Tˆ 2 . For each vertex w ∈ V (G n ), the vertex with prefixing a bit j ∈ {0, 1} in the label of w is denoted by w j . By Theorem 1, we know that there is at least one of T 1 and T 2 such that j it contains w as a leaf. Clearly, if w is a leaf in T i for i ∈ {1, 2}, then so is w j in T i , and this further implies that w j is still a leaf in Tˆ i because Tˆ i is constructed from T i0 and T i1 by adding an edge between two internal vertices of the trees. Thus, this proves the theorem. 2
For the facilitate of discussion, for i ∈ {1, 2}, such vertices u i ∈ V ( T i0 ) and v i ∈ V ( T i1 ) in the above theorem are called
the port vertices of the trees T i0 and T i1 , respectively. Moreover, the edge (u i , v i ) ∈ E (G n+1 ) is called the bridge of Tˆ i for the
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Fig. 9. An illustration of port vertices for pairs {0, 2k } where 4 k n.
construction. Obviously, choosing an appropriate pair {u i , v i } as port vertices in the construction is important because it will determine the diameter of the resulting tree. For example, we consider the construction of two CISTs in Q 5 . Let T 1 and T 2 be two CISTs of the 4-dimensional hypercube constructed in the previous section (see Fig. 1(d)), where T 1 and T 2 take the black vertices as internal vertices and leaves, respectively. Clearly, N 4 (12) = 12 + 24 = 28. If we choose the pair {12, 28} as the port vertices for T 10 and T 11 ,
then both the trees T 10 , T 11 , together with the bridge (12, 28) form a spanning tree Tˆ 1 of Q 5 with diameter 13. Indeed, if we choose vertices near the center of the trees as their port vertices, then we can get a better result of the diameter of CISTs for the construction. Note that the center of a graph is defined to be the set of vertices which minimize the maximal distance from other vertices in the graph. In particular, the cardinality of the center in a tree is at most two. In the case of Q 4 , we would like to choose the pair {0, 16(= 0 + 24 )} as the port vertices for T 10 and T 11 . Similarly, we choose the pair {4, 20(= 4 + 24 )} as the port vertices for T 20 and T 21 . As a result, by Theorem 4, the two CISTs constructed for Q 5 have diameter 9. In general, we construct two CISTs in Q n+1 for n 4 as follows. Suppose that T 1 and T 2 are two CISTs of Q n and let Tˆ 1 and Tˆ 2 be two CISTs of Q n+1 constructed from the rule of Theorem 4. Clearly, for i ∈ {1, 2}, if a vertex u is a center vertex of T i0 , then so is the vertex N n (u ) = u + 2n of T i1 . Thus, the choice of pairs {0, 2n } and {4, 4 + 2n } as port vertices can build two CISTs of Q n+1 by induction on n. See Fig. 9 for an illustration of port vertices in a recursive construction. Since we can always choose center vertices of T 1 and T 2 as port vertices, it follows that
D ( Tˆ i ) = 2 ·
D (T i ) 2
+ 1 for i ∈ {1, 2},
(1)
where D ( T ) denotes the diameter of the tree T . Therefore, by Theorems 3 and 4, we have the following corollary. Corollary 5. For n 4, Q n admits two CISTs with diameter 2n − 1. In what follows, we consider the construction of two CISTs in other high-dimensional hypercube-variant networks. For a 4-dimensional variant hypercube, such as LTQ 4 , CQ 4 , PQ 4 and MQ 4,i for i ∈ {0, 1}, we assume that T 1 and T 2 are two CISTs constructed in the previous section, where T 1 and T 2 take the black vertices as internal vertices and leaves, respectively. For constructing two CISTs in LTQ n+1 with n 4, we first observe the two CISTs of LTQ 4 in Fig. 2(d). It shows that vertices 4 and 5 are in the center of T 1 , and vertices 14 and 15 are in the center of T 2 . Recall that the n-neighbor of a vertex x in LTQ n+1 is defined to be N n (x) = x¯ n (xn−1 ⊕ x0 )xn−2 · · · x0 . Hence, for arbitrary n 4, we have N n (4) = 4 + 2n and N n (14) = 14 + 2n in LTQ n+1 . According to Theorem 4, we can choose the pairs {4, 4 + 2n } and {14, 14 + 2n } as port vertices for building two CISTs of LTQ n+1 by using induction on n. Thus, through an argument similar to the analysis of the diameter of CISTs in hypercubes, we obtain the following corollary. Corollary 6. For n 4, LTQ n admits two CISTs with diameter 2n − 1. For constructing two CISTs in CQ n+1 with n 4, we observe the two CISTs of CQ 4 in Fig. 3(d), and it shows that vertices 0 and 4 are in the center of T 1 , and vertices 3 and 9 are in the center of T 2 . From the definition of the n-neighbor in
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CQ n+1 , we have N n (0) = 2n . Thus, by Theorem 4, we can choose the pair {0, 2n } as port vertices for building the first CIST of CQ n+1 by using induction on n. To build the second CIST of CQ n+1 , unfortunately, if we choose vertex 3 (respectively, vertex 9) as port vertex of a CIST in CQ n0 , then the vertex N n (3) = (3 − 21 ) + 2n (respectively, N n (9) = (9 + 21 ) + 2n ) is a leaf in the corresponding tree of CQ n1 , and thus cannot serve as a port vertex. So, we consider an alternative as shown in Fig. 4, where T 1 is the same as before and T 2 contains only one center vertex 8. Since N n (8) = 8 + 2n , we can choose the pair {8, 8 + 2n } as port vertices for building the second CIST of CQ n+1 . According to Theorems 3, 4, and Eq. (1), we have the following corollary. Corollary 7. For n 4, CQ n admits two CISTs with diameter 2n − 1. For constructing two CISTs in PQ n+1 with n 4, we observe the two CISTs of PQ 4 in Fig. 5(d), and it shows that vertices 14 and 15 are in the center of T 1 , and vertices 2 and 3 are in the center of T 2 . From the definition of the n-neighbor in PQ n+1 , we have N n (14) = (14 − 23 ) + 2n , N n (15) = (15 − 23 − 21 ) + 2n , N n (2) = 2 + 2n , and N n (3) = (3 − 21 ) + 2n . In this case, we can choose the pair {2, 2 + 2n } as port vertices for building the second CIST. However, since both N 4 (14) and N 4 (15) are leaves in T 11 , none of them can serve as port vertex for building the first CIST of PQ 5 . Similar to the situation of crossed cubes, we consider an alternative as shown in Fig. 6. In this construction, it shows that T 1 contains only one center vertex 0 and T 2 contains only one center vertex 2. Since N n (0) = 2n and N n (2) = 2 + 2n , we can choose the pairs {0, 2n } and {2, 2 + 2n } as port vertices for building two CISTs of PQ n by using induction on n. Again, by Theorems 3, 4, and Eq. (1), we obtain the following corollary. Corollary 8. For n 4, PQ n admits two CISTs with diameter 2n − 1. Finally, we consider the construction of two CISTs in MQ n+1 for n 4. Let T 1 and T 2 be two CISTs of MQ 4,0 as shown in Fig. 7(d), where vertices 4 and 7 are in the center of T 1 , and vertices 5 and 13 are in the center of T 2 . Also, let T 1 and T 2 be two CISTs of MQ n,1 as shown in Fig. 8(d), where vertices 4 and 11 are in the center of T 1 , and vertices 2 and 13 are in the center of T 2 . Recall that the n-neighbor of a vertex x in MQ n+1,0 (respectively, in MQ n+1,1 ) is defined n n to be N n (x) = x¯ n xn−1 · · · x0 (respectively, N n (x) = x¯ n x¯ n−1 · · · x¯ 0 ). Hence, n nwe have Nn (4) = 4 + 2 and Nn (13) = 13 + 2 in MQ n+1,0 , and N n (4) = (15 − 4) + k=4 2k and N n (13) = (15 − 13) + k=4 2k in MQ n+1,1 . To construct two CISTs of MQ 5,0 , we choose the pairs {4, 20(= 4 + 24 )} and {13, 29(= 13 + 24 )} as port vertices, where 4 ∈ V ( T 10 ), 20 ∈ V ( T 11 ), 13 ∈ V ( T 20 ), and 29 ∈ V ( T 21 ) are all center vertices in the trees. To construct two CISTs of MQ 5,1 , we choose the pairs {4, 27(= 11 + 24 )} and {13, 18(= 2 + 24 )} as port vertices, where 4 ∈ V ( T 10 ), 27 ∈ V ( T 11 ), 13 ∈ V ( T 20 ), and 18 ∈ V ( T 21 ) are all center vertices in the trees. In general, we can choose the pairs {4, 4 + 2n } and {13, 13 + 2n } as port vertices for building two CISTs of MQ n+1,0 n n by using induction on n. By contrast, we can choose the pairs {4, 11 + k=4 2k } and {13, 2 + k=4 2k } as port vertices for building two CISTs of MQ n+1,1 by using induction on n. As a result, by Theorems 3, 4, and Eq. (1), we obtain the following corollary. Corollary 9. For n 4 and i ∈ {0, 1}, MQ n,i admits two CISTs with diameter 2n − 1. 4. Concluding remarks In this paper, we provide a unified approach for constructing two CISTs in hypercubes and their variants. In addition, we give the exact diameter of the constructed CISTs. Since research results related to CISTs are limited, the following three problems are worthwhile to be addressed in a future research. (1) Is it possible to reduce the diameter of CISTs for hypercube-variant networks mentioned above? (2) How to design algorithms to construct more (e.g. 3) CISTs in high-dimensional hypercube-variant networks? Especially, is there a unified approach? (3) Does there exist a condition of necessity and sufficiency for which Hasunuma’s conjecture is affirmative in a subclass of Q n (or other variants)? References [1] T. Araki, Dirac’s condition for completely independent spanning trees, J. Graph Theory 77 (2014) 171–179. [2] H.-Y. Chang, H.-L. Wang, J.-S. Yang, J.-M. Chang, A note on the degree condition of completely independent spanning trees, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 98-A (2015) 2191–2193. [3] B. Cheng, J. Fan, D. Wang, J. Yang, A reliable broadcasting algorithm in locally twisted cubes, in: Proc. 2nd Int. Conf. on Cyber Security and Cloud Computing, 2015, pp. 323–328. [4] B. Cheng, D. Wang, J. Fan, Constructing completely independent spanning trees in crossed cubes, personal communication. [5] P. Cull, S.L. Larson, The Möbius cubes, IEEE Trans. Comput. 44 (1995) 647–659. [6] K. Efe, A variation on the hypercube with lower diameter, IEEE Trans. Comput. 40 (1991) 1312–1316. [7] K. Efe, The crossed cube architecture for parallel computation, IEEE Trans. Parallel Distrib. Syst. 3 (1992) 513–524.
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