In an altruistic DISH protocol, additional nodes called altruists are deployed in a multi-channel ad hoc network to achieve energy efficiency while still maintaining the original throughput-delay performance. The responsibility of altruists is to constantly monitor the control channel and awaken other (normal) nodes when necessary (to perform data transmissions). Altruists never sleep while other nodes sleep as far as possible. This technical report proves three properties related to this cooperative protocol. The first is the conditions for forming an unsafe pair (UP) in an undirected graph. The second is the necessary and sufficient conditions for full cooperation coverage to achieve the void of multi-channel coordination (MCC) problems. The last is the NP-hardness of determining the minimum number and locations of altruistic nodes to achieve full cooperation coverage.
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