Enumeration and Asymptotic Analysis of Strict Non-Plane Cactus Graphs over a Finite Set of Cycle Lengths

We consider the enumeration of strict cactus graphs, in the free, non-plane setting, whose cyclic blocks are drawn from a finite set of admissible cycle lengths rather than from a single fixed length. For any such set we give an exact combinatorial characterization of which vertex counts occur; for two admissible cycle lengths this characterization takes a closed form via the classical Frobenius coin problem. When every admissible cycle length is odd, we prove a closed-form expression for the critical value governing the exponential growth rate, extending a result previously known only for a single admissible cycle length; when an even cycle length is present, we show that the same method of proof is structurally obstructed. We establish the general shape of the asymptotic enumeration for an arbitrary finite set of admissible cycle lengths, and we obtain a closed form for an associated second-order coefficient in the all-odd case. Numerical data, cross-validated by two independent methods, are given for four representative examples.