Inapproximability of Maximum r-Regular Induced Connected Subgraph Problems

Given a graph G = (V,E) on n vertices, the MAXIMUM r-REGULAR INDUCED CONNECTED SUBGRAPH (r-MaxRICS) problems ask for a maximum sized subset of vertices S ⊆ V such that the induced subgraph G[S] on S is connected and r-regular. For r = 2, it is known that 2-MaxRICS is NP-hard and cannot be approximated within a factor of n1−ε in polynomial time for any ε > 0 if P 6 = NP . In this paper, we show that r-MaxRICS is NP-hard for any fixed integer r ≥ 3, and furthermore rMaxRICS cannot be approximated within a factor of n1/6−ε in polynomial time for any ε > 0 if P 6 = NP .