2026/05/21 by Jianfeng Hou, Hongbin Zhao
Computer Science · Mathematics · #cs.DS #math.CO
14 pages. Comments are welcome
arxiv created 2026/05/21 · arxiv updated 2026/08/05
We study the exact enumeration of maximal cliques in graph classes defined by excluded clique minors and excluded clique immersions. For n-vertex Kt-minor-free graphs, we give an algorithm that lists all maximal cliques in n * 4^(2t/5+o(t)) time, significantly improving the previous n * 2O(t log log t) bound of Eppstein, Löffler, and Strash. For n-vertex Kt-immersion-free graphs, we establish the first exact enumeration algorithm parameterized by immersion number, achieving a running time of n * 3^(t/3+o(t)). While both algorithms employ a common degeneracy-based root-assignment scheme, their analyses require distinct structural mechanisms. Crucially, rather than applying generic sparsity bounds, our algorithms deeply integrate the specific structural obstructions -- local density thresholds for minors and minimum-degree branchings for immersions -- directly into the enumeration logic. We also prove matching output-size lower bounds, up to sub-exponential factors in t, using specialized constructions. Consequently, the exponential bases 4^(2/5) and 3^(1/3) are asymptotically optimal.