Exact capacitated domination: On the computational complexity of uniqueness

Gerke et al. (2019) introduced a game-theoretic model to study public good provision in social networks when there are constraints on sharing. This generates purely graph-theoretic problem termed exact capacitated domination. In the we given graph, graph with parameter defined each vertex that is interpreted as capacity of vertex. The objective find DP-Nash subgraph: spanning bipartite subgraph partite sets D and P, called D-set P-set respectively, such no P isolated adjacent number vertices equal its capacity. We show whether has unique can be decided polynomial time. However, also closely related deciding co-NP-complete.