Reliability Evaluation of Augmented Cubes on Degree

The [Formula: see text]-dimensional augmented cube [Formula: see text], proposed by Choudum and Sunitha in 2002, is one of the most famous interconnection networks of the distributed parallel system. Reliability evaluation of underlying topological structures is vital for fault tolerance analysis of this system. As one of the most extensively studied parameters, the [Formula: see text]-conditional edge-connectivity of a connected graph [Formula: see text], [Formula: see text], is defined as the minimum number of the cardinality of the edge-cut of [Formula: see text], if exists, whose removal disconnects this graph and keeps each component of [Formula: see text] having minimum degree at least [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] be three integers, where [Formula: see text], if [Formula: see text] and [Formula: see text], if [Formula: see text]. In this paper, we determine the exact value of the [Formula: see text]-conditional edge-connectivity of [Formula: see text], [Formula: see text] for each positive integer [Formula: see text] and [Formula: see text], and give an affirmative answer to Shinde and Borse’s corresponding conjecture on this topic in [On edge-fault tolerance in augmented cubes, J. Interconnection Netw. 20(4) (2020), DOI:10.1142/S0219265920500139].

Construction of Quality Virtual Backbones with Link Fault Tolerance in Wireless Sensor Networks

Wireless sensor networks (WSNs) are extensively utilized in various circumstances. For applications, the construction of the virtual backbones (VBs) of WSNs has attracted considerable attention in this field. Generally, a homogeneous WSN is formulated as a unit disk graph (UDG), and the VB of the corresponding WSN is modeled as a connected dominating set (CDS) in the UDG. In certain applications, communication between sensors in a network may fail for various reasons, such as sensor movement, signal interference, and the appearance of obstacles. Consequently, a CDS in a UDG should possess fault tolerance on the edges. In this paper, we introduce a new concept called the 2 edge-connected 2 edge-dominating set ( 2 , 2 -ECDS); then, we design an approximation algorithm for computing 2 , 2 -ECDSs in UDGs, the performance ratio of which is 30.51. By means of simulations, we compare our algorithm and existing algorithms in terms of the CDS size, running time, success rate, and network lifetime. The simulation results indicate that our algorithm exhibits better performance and is more suitable for constructing a VB with edge fault tolerance in a WSN.

On Edge-Fault Tolerance in Augmented Cubes

The augmented cube AQn is one of the important variations of the hypercube Qn. In this paper, we prove that the conditional h-edge connectivity of AQn with n ≥ 3 is 8n − 16 for h = 3 and 2n for h = 2n − 3. We also obtain an upper bound on the conditional h-edge connectivity for odd integer h satisfying [Formula: see text].

Matching Preclusion for Exchanged Hypercubes

As spanning subgraphs of hypercubes, exchanged hypercubes contain less edges but maintain lots of desired properties of hypercubes. This paper considers matching preclusion, a kind of measures of edge-fault tolerance, of exchanged hypercubes EH(s, t). We show that EH(s, t) is maximally matched, that is, for s ≥ t, mp(EH(s, t)) = t + 1 and EH(s, t) is super matched if and only if (s, t) ≠ (1, 1). Comparing with results of matching preclusion for hypercubes, we conclude that exchanged hypercubes maintain the matching preclusion property of hypercubes, except for EH(1, 1).