The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2

A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class parity-P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.

💡 Research Summary

This paper investigates the computational complexity of counting graph homomorphisms modulo 2, focusing on the class of cactus graphs—connected simple graphs in which each edge belongs to at most one cycle. For a fixed target graph H, the problem ⊕HomsTo H asks for the parity (i.e., the count modulo 2) of homomorphisms from an input graph G to H. The authors provide a complete dichotomy: if the involution‑free reduction of H (obtained by repeatedly deleting fixed‑point subgraphs of order‑2 automorphisms) contains at most one vertex, then ⊕HomsTo H can be solved in polynomial time (belongs to FP); otherwise the problem is ⊕P‑complete under polynomial‑time Turing reductions, meaning it is as hard as any problem of computing a Boolean parity of a #P function.

The work builds on the modular counting framework introduced by Papadimitriou and Zachos and the seminal observations of Valiant concerning the dramatic impact of the modulus on complexity. Prior to this study, a dichotomy for the parity counting problem was known only for trees (Faben and Jerrum, 2015), where the involution‑free reduction being a single vertex characterizes tractability. The present paper extends this result to cactus graphs, confirming a conjecture of Faben and Jerrum for this broader class.

Key technical ingredients include:

1. Involution Reduction – An involution (order‑2 automorphism) σ of H induces a subgraph H^σ consisting of σ‑fixed vertices. Lemma 1 (from Faben‑Jerrum) shows that |Hom(G, H)| ≡ |Hom(G, H^σ)| (mod 2) for any G. Repeatedly applying this operation yields a unique involution‑free graph H′, independent of the order of reductions (Lemma 2). Consequently, the parity counting problem for H reduces to that for H′.

2. Hardness Gadgets, Partial Gadgets, and Mosaics – To prove ⊕P‑hardness the authors construct specific substructures within an involution‑free cactus graph. A hardness gadget enables a parsimonious reduction from the parity version of weighted independent set (⊕IS(λ, μ)) to ⊕HomsTo H. When a full gadget does not exist, a partial gadget together with a “mosaic” (a union of 4‑cycles) can be combined recursively to obtain a full hardness gadget. The authors prove that every non‑trivial involution‑free cactus graph contains at least one of these configurations.

3. Pinning Technique – The reduction from ⊕IS(λ, μ) to ⊕HomsTo H proceeds via a pinned homomorphism problem, where a function p: V(G) → 2^{V(H)} forces each vertex of G to map into a prescribed orbit of the automorphism group of H. For asymmetric H (no non‑trivial automorphisms) the orbit is a single vertex, making pinning straightforward. For graphs with non‑trivial automorphisms, the authors augment G with a copy of H and pin each vertex of the copy to its own orbit, ensuring that any homomorphism respecting all orbits must be an automorphism of H. This bridges the pinned problem back to the unpinned ⊕HomsTo H.

4. Automorphism Group Computation – Determining whether H’s involution‑free reduction has ≤1 vertex requires finding an involution and computing |Aut(H)| mod 2. Because cactus graphs are planar and have tree‑width at most 2, both tasks can be performed in polynomial time, making the meta‑problem of classifying H tractable.

The main theorem (Theorem 6) states: for any simple cactus graph H, if the involution‑free reduction of H has at most one vertex then ⊕HomsTo H ∈ FP; otherwise ⊕HomsTo H is ⊕P‑complete. The proof proceeds by (i) showing that any involution‑free cactus graph either already yields a tractable case (empty graph, single vertex with/without a loop, or two isolated vertices with exactly one loop) or (ii) contains a hardness gadget (or can be reduced to one via mosaics), thereby establishing ⊕P‑hardness.

The paper also discusses related work on modular counting for Boolean CSPs, weighted homomorphism sums, and the broader landscape of dichotomy theorems for exact counting (#P) and for other moduli. It emphasizes that, unlike the exact counting setting where no intermediate complexity exists for homomorphisms (the problem is either polynomial‑time solvable or #P‑complete), the parity setting admits a rich structure where the graph’s symmetry properties dictate tractability.

In conclusion, the authors deliver a comprehensive classification of parity homomorphism counting for cactus graphs, confirming the conjectured dichotomy for this class and introducing novel graph‑theoretic constructions (hardness gadgets, mosaics) that may prove useful for extending the result to larger graph families such as bounded‑tree‑width or planar graphs. The work deepens our understanding of how algebraic properties of the target graph interact with combinatorial counting, and it showcases the power of combining group‑theoretic reductions with careful structural decomposition.