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

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.

DISTANCE DOMINATION ON TOPOLOGICAL GRAPHS AND ITS APPLICATIONS

In this paper we introduce topological graph as bus topological graph, ring topological graph, star topological graph, mesh topological graph and hybrid topological graph. We extend the result that if T is a tree and it has maximum degree m then there exist at least m pendant vertices in to if T is a tree except bus topological graph and it has maximum degree m then there exist exactly m pendant vertices.

Exponential Domination Critical and Stability in Some Graphs

An exponential dominating set of graph [Formula: see text] is a kind of distance domination subset [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is the length of a shortest path in [Formula: see text] if such a path exists, and [Formula: see text] otherwise. The minimum exponential domination number, [Formula: see text] is the smallest cardinality of an exponential dominating set. The minimum exponential domination number, [Formula: see text] can be decreased or increased by removal of some vertices from [Formula: see text]. In this paper, we investigate of this phenomenon which is referred to critical and stability in graphs.