Some polynomial classes for the acyclic orientation with parity constraint problem

We study the problem of finding an acyclic orientation of an undirected graph with constrained in-degree parities specified by a subset T of vertices. An orientation is called T -odd if a vertex v has odd in-degree if and only if v P T . While the unconstrained parity orientation problem is polynomial (Chevalier et al. (1983)), imposing acyclicity makes it more challenging, and its complexity remains an open question. Szegedy and Szegedy ( 2006) proposed a randomized polynomial-time algorithm for this problem, but it is not known whether it belongs to co-NP. Furthermore, Gravier et al. (2025) showed the problem becomes NP-complete on partially directed graphs, even when restricted to planar cubic graphs. We identify three necessary conditions for the existence of acyclic T -odd orientation: a global parity condition P, and two conditions S and S ensuring the existence of potential sources and sinks. Following the work of Frank and Kiraly (2002), we define graph classes containing the graphs for which a given subset of the necessary conditions P, S and S is also sufficient for the existence of an acyclic T -odd orientation. We establish the inclusion relationships between these classes. We complete the study of these classes by a characterization of the solvable instances for Cartesian products of paths and cycles. The proofs of these results are all constructive, so that acyclic T -odd orientations can be built in polynomial time whenever they exist. We use these families, along with cliques, to demonstrate the strictness of the class inclusions in our hierarchy.

💡 Research Summary

The paper investigates the “acyclic T‑odd orientation” problem: given an undirected simple graph G and a subset T ⊆ V(G), orient each edge so that every vertex v ∈ T has odd indegree and every vertex v ∉ T has even indegree, while the resulting digraph must be acyclic. The unconstrained version (ignoring acyclicity) is known to be solvable in polynomial time (Chevalier et al., 1983). Adding the global acyclicity constraint makes the problem substantially harder; its exact decision‑complexity remains open. Szegedy and Szegedy (2006) gave a randomized polynomial‑time algorithm, but it is not known whether the problem lies in co‑NP. Moreover, Gravier et al. (2025) proved NP‑completeness for partially directed graphs, even when restricted to planar cubic instances.

To approach the problem, the authors introduce three necessary conditions that any feasible instance must satisfy:

1. Global parity condition P: |E(G)| − |T| must be even. This is the same parity condition that characterizes the unconstrained T‑odd orientation.
2. Potential source condition S: Define Source(T) = {v ∈ V\T | deg(v) ≡ 1 (mod 2)}. In any acyclic orientation there must be at least one vertex of indegree 0, which forces Source(T) to be non‑empty (unless the graph is a singleton). Moreover, if |Source(T)| = 1 then either the graph has a single vertex or Source(T) and Sink(T) must be distinct.
3. Potential sink condition S (the symmetric counterpart): Define Sink(T) = {v ∈ T | deg(v) ≡ 1 (mod 2)}. An acyclic orientation must also contain a vertex of outdegree 0, so Sink(T) must be non‑empty under analogous restrictions.

The three conditions are denoted P, S, and S. The authors then consider all subsets N ⊆ {P,S,S} and define a class C_N as the set of graphs for which the conditions in N are not only necessary but also sufficient: for every T satisfying the conditions in N, the graph admits an acyclic T‑odd orientation. This yields seven classes: C_P, C_S, C_S, C_PS, C_PS, C_SS, and C_PSS. Inclusion relationships are immediate: if N₁ ⊆ N₂, then C_{N₁} ⊆ C_{N₂}.

Theorem 1.1 provides a complete characterization of these classes:

• (a) C_S = C_S and (b) C_SS = C_SS (the two source‑only and sink‑only classes coincide with their symmetric versions).
• (c) C_P consists exactly of connected, non‑Eulerian graphs (i.e., at least one vertex has odd degree) with |V| − |E| ≡ 1 (mod 2).
• (d) C_PS = C_P ∪ C_PS; equivalently, a graph belongs to C_PS iff it is either in C_P or satisfies both P and the source condition.
• (e) C_PSS consists precisely of connected, non‑Eulerian graphs with |V| − |E| ≡ 0 (mod 2).

Thus, the global parity condition together with source/sink requirements fully determines feasibility for several natural families of graphs.

The paper also revisits a result of Király and Kisfaludi‑Bak (Theorem 1.2) which states that a graph belongs to C_P iff it admits an acyclic T₁‑odd orientation for some T₁ that is the complement of a single vertex. This highlights the importance of a “potential source” vertex.

The second major contribution concerns Cartesian products of paths and cycles, i.e., grids, cylinders, and tori. The authors completely characterize when such products admit an acyclic T‑odd orientation for any T satisfying the three necessary conditions.

• Theorem 1.3: Let G = C_p ⊠ H where C_p is a cycle on p vertices and H is either a path or a cycle on q vertices. If H is a path, or if both p and q are at least 4, then G always admits an acyclic T‑odd orientation for any T satisfying P, S, S. The proof is constructive: one can orient edges along the cycle and path dimensions while preserving acyclicity and the required parity.
• Theorem 1.4: For the grid P_p ⊠ P_q (the Cartesian product of two paths), an acyclic T‑odd orientation exists unless both p and q are even and T obeys a very specific parity pattern: for every even‑indexed row i, the set of columns j with (v_i, w_j) ∈ T must be exactly the odd‑indexed columns, and symmetrically for columns. In all other cases, a polynomial‑time construction yields a feasible orientation.

These results show that for most product graphs the necessary conditions are also sufficient, with the grid being the only family where a non‑trivial additional restriction appears.

Corollary 1.5 demonstrates that the inclusions among the classes are strict. Using cylinders, grids, tori, and cliques, the authors exhibit explicit families that belong to one class but not to a strictly larger one:

• Cylinders C_{2p+1} ⊠ P_{2q} and grids P_{2p+1} ⊠ P_q lie in C_P but not in C_PS.
• Tori C_p ⊠ C_q (p,q ≥ 4) and simple cycles belong to C_PS ⊂ C_P.
• Cliques K_{4k+2} are in C_P but not in C_PS; cliques of the form K_{2k+1} ⊠ P_t are in C_PS but not in C_PSS, etc.

These constructions confirm that each additional condition genuinely narrows the class of graphs for which the parity constraints become sufficient.

All proofs are constructive: given a graph G and a set T that meets the relevant conditions, the authors present explicit polynomial‑time algorithms to produce an acyclic T‑odd orientation. Consequently, the paper not only advances the theoretical understanding of the problem’s structure but also provides practical tools for algorithmic implementation.

In summary, the work establishes a clear hierarchy of graph classes based on necessary parity, source, and sink conditions, fully characterizes these classes for several important families (especially Cartesian products of paths and cycles), and proves that the hierarchy is strict. While the overall decision problem remains open in terms of complexity classification, the constructive results and the identified polynomial‑time solvable subclasses represent a significant step forward in the study of parity‑constrained acyclic orientations.