An Improved Nordhaus–Gaddum-Type Theorem for 2-Rainbow Independent Domination Number

For a graph G, its k-rainbow independent domination number, written as γrik(G), is defined the cardinality of minimum set consisting k vertex-disjoint sets V1,V2,…,Vk such that every vertex in V0=V(G)\(∪i=1kVi) has neighbor Vi for all i∈{1,2,…,k}. This invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics Computation 333(15), 2018: 353–361), which aims to compute number G□Kk (the generalized prism) via studying problem integer labeling on G. They proved Nordhaus–Gaddum-type theorem: 5≤γri2(G)+γri2(G¯)≤n+3 any n-order G with n≥3, G¯ denotes complement work improves their result shows if G≇C5, then 5≤γri2(G)+γri2(G¯)≤n+2.