Optimization Models for Efficient (t, r) Broadcast Domination in Graphs

Poompol Buathong, Tipaluck Krityakierne

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.