Counting Subgraphs in Degenerate Graphs

We consider the problem of counting number copies a fixed graph H within an input G . This is one most well-studied algorithmic problems, with many theoretical and practical applications. focus on solving this when has bounded degeneracy rich family graphs, containing all graphs without minor (e.g., planar graphs), as well generated by various random processes preferential attachment graphs). say that easy if there linear-time algorithm for in degeneracy. A seminal result Chiba Nishizeki from ’85 states every at 4 vertices easy. Bera, Pashanasangi, Seshadhri recently extended to 5 further proved \( k \gt \) -vertex which not They left open natural characterizing Bressan introduced framework subgraphs degenerate can extract sufficient condition be Here, we show also necessary, thus fully answering Bera–Pashanasangi–Seshadhri problem. resolve two closely related problems; namely are respect induced copies, homomorphisms.