Odd and Co – Odd Sum Degree Domination in Graphs with Algorithm

• Published in: Indian journal of science and technology 2024-08, Vol.17 (32), p.3368-3376
• Main Authors: Ilakkiya, D, Selvi, V Mohana
• Format: Article
• Language: English
• ISSN: 0974-6846; 0974-5645
• DOI: 10.17485/IJST/v17i32.1295

Objectives: The main aim of this study is to introduce two new domination parameters, namely odd and co-odd sum degree domination numbers, and to provide an algorithm for finding the odd sum degree dominating set of a graph. Methods: The odd and co-odd sum degree domination numbers of a graph G are introduced by verifying the following two conditions on dominating and co-dominating sets of G. (i) The sum of degree of vertices of the dominating and co-dominating sets are odd numbers. (ii) The cardinality of the above two sets are minimum. Further, obtained the degree sum values of the above two sets are referred as odd and co-odd sum degree domination values of G. Findings: The odd and co-odd sum degree domination numbers and values are defined and their exact values are obtained for some standard classes of graphs. The bounds of the above numbers are found in terms of basic graph theoretical terminologies. Further, the above said domination parameters are characterized. The relationships with other domination parameters are determined. Novelty: Domination is a wide area with more applications in real-life situations. It has numerous applications in modern science and engineering. The odd and even numbers play a vital role in the following areas like Number Theory in Mathematics, Sorting Algorithms in Computer Science, Arrangements of atoms in Molecular Chemistry, etc. The degree-based domination also has plenty of applications like ensuring redundancy and reliability in Communication Networks, identifying sets of nodes or links that provide fault tolerance in Electrical Networks, etc. Motivated by the notion of the above applications in this paper, the odd and co-odd sum degree dominations are introduced. Further, the step-by-step algorithm for finding the newly introduced parameter is also presented with an illustration. Keywords: Odd sum degree dominating set, Odd sum degree domination number, Odd sum degree domination value, Co-odd sum degree dominating set, Co-odd sum degree domination number, and Co-odd sum degree domination value.