Augmented Cube Research Articles

Longest Cycles Embedding in Faulty Augmented Cube Networks

The augmented cube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. In this paper, the topological structure of dimensional augmented cube is analyzed. The properties related to cycle embedding in faulty have been investigated. Let denote the set of fault links. This study demonstrates that when , for each nonfault link of augmented cube networks, lies on a longest fault free cycle which contains every nodes of .

Augmented k-ary n-cubes

We define an interconnection network AQ n, k which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQ n, k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube AQ n, k : is a Cayley graph, and so is vertex-symmetric, but not edge-symmetric unless n = 2; has connectivity 4 n − 2 and wide-diameter at most max{( n − 1) k − ( n − 2), k + 7}; has diameter k 3 + k - 1 3 , when n = 2; and has diameter at most k 4 ( n + 1 ) , for n ⩾ 3 and k even, and at most k 4 ( n + 1 ) + n 4 , for n ⩾ 3 and k odd.

The panpositionable panconnectedness of augmented cubes

A graph G is panconnected if, for any two distinct vertices x and y of G, it contains an [ x, y]-path of length l for each integer l satisfying d G ( x, y) ⩽ l ⩽ ∣ V( G)∣ − 1, where d G ( x, y) denotes the distance between vertices x and y in G, and V( G) denotes the vertex set of G. For insight into the concept of panconnectedness, we propose a more refined property, namely panpositionable panconnectedness. Let x, y, and z be any three distinct vertices in a graph G. Then G is said to be panpositionably panconnected if for any d G ( x, z) ⩽ l 1 ⩽ ∣ V( G)∣ − d G ( y, z) − 1, it contains a path P such that x is the beginning vertex of P, z is the ( l 1 + 1)th vertex of P, and y is the ( l 1 + l 2 + 1)th vertex of P for any integer l 2 satisfying d G ( y, z) ⩽ l 2 ⩽ ∣ V( G)∣ − l 1 − 1. The augmented cube, proposed by Choudum and Sunitha [6] to be an enhancement of the n-cube Q n , not only retains some attractive characteristics of Q n but also possesses many distinguishing properties of which Q n lacks. In this paper, we investigate the panpositionable panconnectedness with respect to the class of augmented cubes. As a consequence, many topological properties related to cycle and path embedding in augmented cubes, such as pancyclicity, panconnectedness, and panpositionable Hamiltonicity, can be drawn from our results.

Edge-fault-tolerant vertex-pancyclicity of augmented cubes

The n-dimensional augmented cube, denoted as AQ n , a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQ n lies on a fault-free cycle of every length from 3 to 2 n , even if there are up to n − 1 edge faults. We also show that our result is optimal.

Conditional edge-fault Hamiltonicity of augmented cubes

The augmented cube is a variation of hypercubes, it possesses many superior properties. In this paper, we show that, for any n-dimensional augmented cube ( n ⩾ 3 ) with faulty edges up to 4 n - 8 in which each vertex is incident to at least two fault-free edges, there exists a fault-free Hamiltonian cycle. Our result is optimal with respect to the number of faulty edges tolerated.

MATCHING PRECLUSION AND CONDITIONAL MATCHING PRECLUSION FOR AUGMENTED CUBES

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the augmented cubes, a class of networks designed as an improvement of the hypercubes.

Fault-tolerant panconnectivity of augmented cubes

The augmented cube AQn is a variation of the hypercube Qn. This paper considers the panconnectivity of AQn (n 3) with at most 2n− 5 faulty vertices and/or edges and shows that, for any two fault-free vertices u and v with distance d in AQn, there exist fault-free uv-paths of every length from d +2 to 2 n − f − 1, where f is the number of faulty vertices in AQn. The proof is based on an inductive construction.

Embedding Hamiltonian paths in augmented cubes with a required vertex in a fixed position

It is proved that there exists a path P l ( x , y ) of length l if d A Q n ( x , y ) ≤ l ≤ 2 n − 1 between any two distinct vertices x and y of A Q n . Obviously, we expect that such a path P l ( x , y ) can be further extended by including the vertices not in P l ( x , y ) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that there exists a hamiltonian path R ( x , y , z ; l ) from x to z such that d R ( x , y , z ; l ) ( x , y ) = l for any three distinct vertices x , y , and z of A Q n with n ≥ 2 and for any d A Q n ( x , y ) ≤ l ≤ 2 n − 1 − d A Q n ( y , z ) . Furthermore, there exists a hamiltonian cycle S ( x , y ; l ) such that d S ( x , y ; l ) ( x , y ) = l for any two distinct vertices x and y and for any d A Q n ( x , y ) ≤ l ≤ 2 n − 1 .

A note on “The super connectivity of augmented cubes”

The aim of this note is to mend a flaw in the proof of Theorem 2 in our paper [M. Ma, G. Liu, J.-M. Xu, The super connectivity of augmented cubes, Information Processing Letters 106 (2008) 59–63].

Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes

Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2002) 71–84] proposed the class of augmented cubes as a variation of hypercubes and showed that augmented cubes possess several embedding properties that the hypercubes and other variations do not possess. Recently, Hsu et al. [H.-C. Hsu, P.-L. Lai, C.-H. Tsai, Geodesic-pancyclicity and balanced pancyclicity of augmented cubes, Information Processing Letters 101 (2007) 227–232] showed that the n-dimensional augmented cube AQ n , n ⩾ 2, is weakly geodesic-pancyclic, i.e., for each pair of vertices u , v ∈ AQ n and for each integer ℓ satisfying max { 2 d ( u , v ) , 3 } ⩽ ℓ ⩽ 2 n where d( u, v) denotes the distance between u and v in AQ n , there is a cycle of length ℓ that contains a u- v geodesic. In this paper, we obtain a stronger result by proving that AQ n , n ⩾ 2, is indeed geodesic-pancyclic, i.e., for each pair of vertices u , v ∈ AQ n and for each integer ℓ satisfying max { 2 d ( u , v ) , 3 } ⩽ ℓ ⩽ 2 n , every u- v geodesic lies on a cycle of length ℓ. To achieve the result, we first show that AQ n - f , n ⩾ 3, remains panconnected (and thus is also edge-pancyclic) if f ∈ AQ n is any faulty vertex. The result of fault-tolerant panconnectivity is the best possible in the sense that the number of faulty vertices in AQ n , n ⩾ 3, cannot be increased.

Automorphisms of augmented cubes

A variation of the hypercube, the augmented cube AQ n of dimension n is defined as follows. It has 2 n vertices, each labelled by an n-bit binary string a 1 a 2···a n . Define AQ 1=K 2. For n≥2, AQ n is obtained by taking two copies and of AQ n−1, with vertex sets , , and joining 0 a 2 a 3···a n with 1 b 2 b 3···b n iff either (i) a 2 a 3···a n =b 2 b 3···b n , or (ii) . In this paper, we observe that AQ n is a Cayley graph and identify its automorphism group.

The super connectivity of augmented cubes

The augmented cube AQ n , proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84], is a ( 2 n − 1 ) -regular ( 2 n − 1 ) -connected graph ( n ≠ 3 ). This paper determines that the super connectivity of AQ n is 4 n − 8 for n ⩾ 6 and the super edge-connectivity is 4 n − 4 for n ⩾ 5 . That is, for n ⩾ 6 (respectively, n ⩾ 5 ), at least 4 n − 8 vertices (respectively, 4 n − 4 edges) of AQ n are removed to get a disconnected graph that contains no isolated vertices. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.

The distinguishing number of the augmented cube and hypercube powers

The distinguishing number of a graph G , denoted D ( G ) , is the minimum number of colors such that there exists a coloring of the vertices of G where no nontrivial graph automorphism is color-preserving. In this paper, we answer an open question posed in Bogstad and Cowen [The distinguishing number of the hypercube, Discrete Math. 283 (2004) 29–35] by showing that the distinguishing number of Q n p , the p th graph power of the n -dimensional hypercube, is 2 whenever 2 < p < n - 1 . This completes the study of the distinguishing number of hypercube powers. We also compute the distinguishing number of the augmented cube AQ n , a variant of the hypercube introduced in Choudum and Sunitha [Augmented cubes, Networks 40 (2002) 71–84]. We show that D ( AQ 1 ) = 2 ; D ( AQ 2 ) = 4 ; D ( AQ 3 ) = 3 ; and D ( AQ n ) = 2 for n ⩾ 4 . The sequence of distinguishing numbers { D ( AQ n ) } n = 1 ∞ answers a question raised in Albertson and Collins [An introduction to symmetry breaking in graphs, Graph Theory Notes N.Y. 30 (1996) 6–7].

Cycle embedding of augmented cubes

A graph G = ( V , E ) is vertex-pancyclic if for every vertex u and any integer l ranging from a positive constant L to | V | , G contains a cycle C of length l such that u is in C. In this paper, we show that an n-dimensional augmented cube, where n ⩾ 2 , is vertex-pancyclic with L = 3 .

Fault-tolerant pancyclicity of augmented cubes

As an enhancement on the hypercube Q n , the augmented cube AQ n , prosed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84], not only retains some favorable properties of Q n but also possesses some embedding properties that Q n does not. For example, AQ n is pancyclic, that is, AQ n contains cycles of arbitrary length for n ⩾ 2 . This paper shows that AQ n remains pancyclic provided faulty vertices and/or edges do not exceed 2 n − 3 and n ⩾ 4 .

Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes

As an enhancement on the hypercube Q n , the augmented cube AQ n , proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks, 40(2) (2002), 71–84], not only retains some of the favorable properties of Q n but also possesses some embedding properties that Q n does not. For example, AQ n contains cycles of all lengths from 3 to 2 n , but Q n contains only even cycles. In this paper, we obtain two stronger results by proving that AQ n contains paths, between any two distinct vertices, of all lengths from their distance to 2 n − 1; and AQ n still contains cycles of all lengths from 3 to 2 n when any (2 n − 3) edges are removed from AQ n . The latter is optimal since AQ n is (2 n − 1)-regular.

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u , v ∈ V ( G ) , a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if d C ( u , v ) = max { d C ( x , y ) | x , y ∈ V ( C ) } . A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271–278] if it contains a cycle of every length from 3 to | V ( G ) | inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181–187] (respectively, balanced pancyclic) if for each pair of vertices u , v ∈ V ( G ) , it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max { 2 d G ( u , v ) , 3 } ⩽ l ⩽ | V ( G ) | . Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251] proved that the n-dimensional Augmented cube, AQ n , is pancyclic in the sense that a cycle of length l exists, 3 ⩽ l ⩽ | V ( AQ n ) | . In this paper, we study two new pancyclic properties and show that AQ n is geodesic pancyclic and balanced pancyclic for n ⩾ 2 .

A reexamination of the size–weight illusion induced by visual size cues

The size-weight illusion induced by visually perceived sizes was reexamined to investigate whether this illusion is a sensory based or cognitive-based phenomenon. A computer-augmented environment was utilized to manipulate visual size information of target objects independently of their haptic information. Two physical cubes of equal mass (30.0 g) and size (3.0 x 3.0 x 3.0 cm) were suspended in parallel by wires attached to small graspable rings, in order to keep haptically obtained information constant between lifts. Instead of directly seeing each physical cube, subjects viewed 3D graphics of a cube with a wire and a ring that were precisely superimposed onto each physical cube. Seventeen subjects vertically lifted these augmented cubes, one after the other, by grasping the attached rings, and then reported their perception of cube heaviness. The graphical size of a comparison cube pseudo randomly varied for every comparison from 1.0 x 1.0 x 1.0 to 9.0 x 9.0 x 9.0 cm, while that of a standard cube remained constant (5.0 x 5.0 x 5.0 cm). Results indicated that the size-weight illusion frequently and systematically occurred for all the subjects such that when the comparison cube was relatively smaller than the standard cube, it was perceived to be heavier, and vice versa. As the size difference increased between the standard cube and the comparison cube, more subjects experienced the illusion, and vice versa. Follow-up tests showed occurrence of the size-weight illusion was significantly correlated with subject's sensitivity to discriminate weight, but not with sensitivity to discriminate visual size. Results suggest that the size-weight illusion induced by only visual size cues in an augmented environment is sensory based, and depends on an individual's integrated perception based on multimodal sensory information.

The forwarding indices of augmented cubes

For a given connected graph G of order n, a routing R in G is a set of n ( n − 1 ) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of G is the maximum number of paths in R passing through any vertex (resp. edge) in G. Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2002) 71–84] proposed a variant of the hypercube Q n , called the augmented cube AQ n and presented a minimal routing algorithm. This paper determines the vertex and the edge forwarding indices of AQ n as 2 n 9 + ( − 1 ) n + 1 9 + n 2 n 3 − 2 n + 1 and 2 n − 1 , respectively, which shows that the above algorithm is optimal in view of maximizing the network capacity.

Fault hamiltonicity of augmented cubes

In this paper, we consider the fault hamiltonicity and the fault hamiltonian connectivity of the augmented cubes AQ n . Assume that F ⊆ V( AQ n ) ∪ E( AQ n ) and n ⩾ 4. We prove that AQ n − F is hamiltonian if ∣ F∣ ⩽ 2 n − 3 and that AQ n − F is hamiltonian connected if ∣ F∣ ⩽ 2 n − 4. Moreover, these bounds are tight.