The study reported herein attempts to understand the neural mechanisms engaged in the conscious control of breathing and breath-hold. The variations in the electroencephalogram (EEG) based functional connectivity (FC) of the human brain during consciously controlled breathing at 2 cycles per minute (cpm), and breath-hold have been investigated and reported here. An experimental protocol involving controlled breathing and breath-hold sessions, synchronized to a visual metronome, was designed and administered to 20 healthy subjects (9 females and 11 males). EEG data were collected during these sessions using the 61-channel eego mylab system from ANT Neuro. Further, FC was estimated for all possible pairs of EEG time series data, for 7 EEG bands. Feature selection using a genetic algorithm (GA) was performed to identify a subset of functional connections that would best distinguish the inhale, exhale, inhale-hold, and exhale-hold phases using a random committee classifier. The best accuracy of 93.36 % was obtained when 1161 theta-band functional connections were fed as input to the classifier, highlighting the efficacy of the theta-band functional connectome in distinguishing these phases of the respiratory cycle. This functional network was further characterized using graph measures, and observations illustrated a statistically significant difference in the efficiency of information exchange through the network during different respiratory phases.