Relay coordination is very important from the point of view of security and healthy operation of power system. Relays should be organized in such a way that every relay should have at least one back up relay and coordination time interval between primary and secondary relays should be maintained. This paper presents the solution of relay coordination problem using Teaching learning based optimization technique (TLBO) by utilizing an IEEE 5 bus and IEEE 30 bus test system. The obtained results are compared with the results obtained from other contemporary meta-heuristic techniques like Genetic Algorithm (GA) and Particle Swarm Optimization technique (PSO). It is observed that TLBO is the best among those with respect to process speed. After this we have done an extension of the work by introducing critical element and network graph theory based relay minimization technique. It is expected that this analysis will be helpful to maintain the network reliability and healthy flow of power.