Some polynomial classes for the acyclic orientation with parity constraint problem
This work addresses a theoretical graph theory problem with potential applications in network design, but it is incremental as it builds on prior complexity studies and focuses on specific graph families.
The paper tackles the problem of finding acyclic orientations in undirected graphs with parity constraints on in-degrees, identifying necessary conditions and characterizing solvable instances for specific graph classes, enabling polynomial-time construction of such orientations when they exist.
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.