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# Construction of Minimum Power 3-Connected Subgraph with k Backbone Nodes in Wireless Sensor Networks

• D. Pushparaj Shetty
• M. Prasanna Lakshmi
Conference paper
Part of the Trends in Mathematics book series (TM)

## Abstract

Minimizing the total power in a wireless sensor network (WSN) has great significance, since the nodes are powered by a small battery of limited capacity. By using an appropriate topology, the energy utilization of the network can be minimized which results in an increased lifetime of a WSN. In reality, WSN is modeled as an undirected graph in which each vertex represents a sensor node and an edge represents the link between the two sensor nodes. We define a distance function that maps a pair of vertices to a positive real number, i.e., Euclidean distance between the two vertices. On this initial topology, we construct a reduced topology satisfying special connectivity constraints like bi-connectivity, k-connectivity, bounded diameter, degree restricted, etc. We assign power to each node as the maximum distance of all its adjacent edges, and total power of the network is the sum of the powers of all the vertices. Fault tolerance addresses the issue of a node or link failure in a WSN. Fault-tolerant network aims at k-connectivity in the network so that there exist at least k vertex disjoint paths between any two sensor nodes of the network. Minimum power 2-connected subgraph (MP2CS) problem is to contrive a 2-connected network with minimum total power. It is proved that MP2CS problem is NP-hard. Minimum power k backbone node 2-connected subgraph (MPkB2CS) problem is a special case of MP2CS problem, which seeks a power assignment satisfying 2-connectivity with k backbone nodes. In this paper, the problem of finding a 3-connected network for a given set of nodes, which minimizes the total power with k backbone nodes, is addressed which is termed as MPkB3CS problem. We propose an algorithm for MPkB3CS problem and establish that the proposed algorithm has an approximation ratio of 4k + 1, for k ≥ 3.

## Keywords

Wireless sensor networks Graph algorithms Topology control problem Range assignment Approximation algorithm

## Notes

### Acknowledgements

The authors would like to acknowledge the National Institute of Technology Karnataka, Surathkal, for the support in this research work.

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.National Institute of Technology KarnatakaMangaluruIndia