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A coevolution model by coupling mortality and fertility selection is introduced to investigate the evolution of cooperation and network structure in the prisoner's dilemma game. The cooperation level goes through a continuous phase transition vs. defection temptation b for low mortality selection intensity β and through a discontinuous one for infinite β. The cooperation level is enhanced most at β≈ 1 for any b. The local and global properties of the network structure, such as cluster and cooperating k-core, are investigated for the understanding of cooperation evolution. Cooperation is promoted by forming a tight cooperating k-core at moderate β, but too large β will destroy the cooperating k-core rapidly resulting in a rapid drop of the cooperation level. Importantly, the infinite β changes the normalized sucker's payoff S from 0 to 1− b and its dynamics of the cooperation level undergoes a very slow power-law decay …