Abstract

Known to be NP-complete, domination number problems in graphs and networks arise in many real-life applications, ranging from the design of wireless sensor networks and biological networks to social networks. Initially introduced by Blessing et al., the (t,r) broadcast domination number is a generalization of the distance domination number. While some theoretical approaches have been addressed for small values of t,r in the literature; in this work, we propose an approach from an optimization point of view. First, the (t,r) broadcast domination number is formulated and solved using linear programming. The efficient broadcast, whose wasted signals are minimized, is then found by a genetic algorithm modified for a binary encoding. The developed method is illustrated with several grid graphs: regular, slant, and king’s grid graphs. The obtained computational results show that the method is able to find the exact (t,r) broadcast domination number, and locate an efficient broadcasting configuration for larger values of t,r than what can be provided from a theoretical basis. The proposed optimization approach thus helps overcome the limitations of existing theoretical approaches in graph theory.

Highlights

  • Consider a connected graph G = (V ( G ), E( G )) with a vertex set V ( G ) and an edge setE( G )

  • We present the results of our studies on the linear programming problem for the (t, r ) broadcast domination number, followed by results of the γ(t, r )

  • Efficient broadcast via a genetic algorithm modified for binary encoding

Read more

Summary

Introduction

A dominating set of the graph G is defined as a subset of vertices D ⊆ V ( G ) such that every vertex v ∈ V ( G ) \ D is adjacent to at least one vertex in D. The domination number of the graph G, denoted by γ( G ), is the minimum size of dominating sets. A variant of broadcast domination depending on two integer parameters t and r was introduced by Blessing et al [1]. This class of domination is called the (t, r ) broadcast domination. D(u, v) can be defined as the number of edges in a shortest path connecting u and v. Let us state required definitions: published maps and institutional affiliations

Objectives
Results
Conclusion

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.