  Network structure is a mechanism for promoting cooperation in social dilemma
games. In the present study, we explore graph surgery, i.e., to slightly
perturb the given network, towards a network that better fosters cooperation.
To this end, we develop a perturbation theory to assess the change in the
propensity of cooperation when we add or remove a single edge to/from the given
network. Our perturbation theory is for a previously proposed random-walk-based
theory that provides the threshold benefit-to-cost ratio, $(b/c)^*$, which is
the value of the benefit-to-cost ratio in the donation game above which the
cooperator is more likely to fixate than in a control case, for any finite
networks. We find that $(b/c)^*$ decreases when we remove a single edge in a
majority of cases and that our perturbation theory captures at a reasonable
accuracy which edge removal makes $(b/c)^*$ small to facilitate cooperation. In
contrast, $(b/c)^*$ tends to increase when we add an edge, and the perturbation
theory is not good at predicting the edge addition that changes $(b/c)^*$ by a
large amount. Our perturbation theory significantly reduces the computational
complexity for calculating the outcome of graph surgery.
