Vol.:(0123456789) 1 3 Journal of Ambient Intelligence and Humanized Computing https://doi.org/10.1007/s12652-020-01740-6 ORIGINAL RESEARCH A study on domatic number of cycle related graphs A. Antony Mary 1  · A. Amutha 2 Received: 15 October 2019 / Accepted: 27 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract A domatic partition of G is the partition of vertices V (G) into disjoint dominating sets. The maximum size of disjoint domi- nating sets is called the domatic number of G. In this paper, comparative results are investigated on domatic partition of few graphs of cycle related graphs such as complete graph, tadpole graph, lollipop graph and barbell graph for cognitive wireless sensor networks. Then the middle graph and central graph of these graphs are studied and domatic number of the defned graphs are determined to fnd the nodes in disjoint sets to disribute the tasks uniformly rather than burden the nodes in domatic set. Furthermore, diameter and domination number of these graphs are observed. Keywords Diameter · Diametrically uniform · Tadpole graph · Lollipop graph · Barbell graph · Dominat set · Cognitive WSN 1 Intoduction Domination is a well known concept of graph theory because of its genuine real life applications in several felds like analysis of design and communication networks, opti- mization, coding theory, etc. The domination concept exists in all kinds of problems. Here, our problem of interest is the domatic number problem (Cockayne and Hedetnieme 1977). The problem of fnding the domatic number is NP-complete for general graphs. There are numerous applications and results on this problem. One of the important applications is the communication networks (Kaplan and Shamir 1994). The network is demonstrated by a graph in which vertices and edges represents cities and communication links. Set of cities together considered as a transmitting group that performs as transmitting stations, can send messages to all cities in the network and this transmitting group in the net- work is a dominating set in the graph. This promotes to fnd a domatic partition in the corresponding graph. Inspired by this reality, we would like to concentrate on cycle related graphs. Since cycle related graphs are signifcant focal point of consideration. The dominating set has been studied elaborately (Berge 1973; Ore 1962). In the present paper some of the important results are discussed and observed on cycle related graphs. First we discussed domatic partition of few graphs of cycle graphs. Then domatic number of complete graph, tadpole graph, lollipop graph and barbell graph is explored. Then, the central and middle graph of these graphs is studied elab- orately. Recent research on central graphs have been studied (Vernold 2007).The idea of middle graph was executed to extend the work (Hamada et al. 1976). The main objective is to fnd the domatic number of central and middle graph of the defned graphs. Also, diameter and domination number of G is focused which is an important graph theoretic param- eter and comparative results are observed. The dominating set nodes in wireless sensor networks have to be tasked in distribute nature. The data sensing and gathering tasks in WSN should be in cognitive nature to improve the network lifetime of such nodes (Idoudi et al. 2019). Since, these nodes share the tasks and other nodes can go into a energy-efcient sleep mode to share the work load (Johny and Meenakshi 2019). The subset of nodes engaged in such sensing takes lose energy quickly because they are busy all the time for sensing, processing, and transmitting data (Islam et al. 2009). Among multiple solutions, the optimal way to distribute the tasks and save * A. Antony Mary anto.doss@yahoo.com A. Amutha amuthajo@gmail.com 1 Department of Mathematics, Sathyabama Institute of Science and Technology, Chennai, India 2 Department of Mathematics, The American College, Madurai, India