Sylvain Gravier

Laurent Beaudou, Pierre Coupechoux, Antoine Dailly et al.

Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, an octal game on a path P\_n is equivalent to playing the same octal game on a heap of n counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars.

Sylvain Gravier, Matthieu Petiteau, Isabelle Sivignon

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.