  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.
