Essentially Tight Kernels for (Weakly) Closed Graphs

Abstract We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider recently introduced parameters number c and weak $$\gamma $$ ? (Fox et al. SIAM J Comput 49(2):448–464, 2020) in addition to standard parameter solution size k . The a is upper-bounded by minimum its degeneracy d For Capacitated Vertex Cover , Connected Induced Matching obtain first kernels $$k^{\mathcal {O}(\gamma )}$$ k O ( ) $$(\gamma k)^{\mathcal respectively. This extends previous results on these degenerate graphs. These are essentially tight as unlikely admit $$k^{o(\gamma o their complexity (Cygan ACM Trans Algorithms 13(3):43:1–43:22, 2017). show that even kernel $$k^{o(c)}$$ c unlikely. In contrast, for with $$\mathcal {O}(ck^2)$$ 2 vertices. Moreover, prove searching an induced subgraph order at least belonging hereditary class {G}$$ G admits contains all complete edgeless Finally, provide lower bounds Independent Set constant Dominating weakly closed split bipartite