Set graphs. II. Complexity of set graph recognition and similar problems
A graph $G$ is said to be a set graph' if it admits an acyclic orientation that is also extensional’, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set. In this paper, we continue the study of set graphs and related topics, focusing on computational complexity aspects. We prove that set graph recognition is NP-complete, even when the input is restricted to bipartite graphs with exactly two leaves. The problem remains NP-complete if, in addition, we require that the extensional acyclic orientation be also slim', that is, that the digraph obtained by removing any arc from it is not extensional. We also show that the counting variants of the above problems are #P-complete, and prove similar complexity results for problems related to a generalization of extensional acyclic digraphs, the so-called hyper-extensional digraphs’, which were proposed by Aczel to describe hypersets. Our proofs are based on reductions from variants of the Hamiltonian Path problem. We also consider a variant of the well-known notion of a separating code in a digraph, the so-called `open-out-separating code’, and show that it is NP-complete to determine whether an input extensional acyclic digraph contains an open-out-separating code of given size.
💡 Research Summary
The paper investigates the computational complexity of recognizing set graphs and several closely related problems. A set graph is defined as an undirected graph that admits an acyclic orientation in which the out‑neighbourhoods of all vertices are pairwise distinct; equivalently, it is the underlying graph of the digraph representation of a hereditarily finite set. The authors focus on decision, counting, and code‑theoretic variants of this recognition problem.
The first major result shows that the set‑graph recognition problem is NP‑complete. The reduction is from a variant of the Hamiltonian Path problem. Importantly, the hardness holds even when the input graph is restricted to be bipartite and to have exactly two leaves. This demonstrates that the difficulty does not stem from complex graph topology but is inherent to the extensional‑acyclic orientation requirement itself.
A second contribution introduces the notion of a “slim” extensional acyclic orientation: an orientation that loses extensionality as soon as any single arc is removed. The authors prove that requiring the orientation to be slim does not lower the complexity; the slim‑extensional recognition problem remains NP‑complete, again via a Hamiltonian‑path‑based reduction that forces the orientation to be minimal.
The paper then turns to counting versions. Determining the number of distinct extensional acyclic orientations of a given graph is shown to be #P‑complete. This aligns with the standard paradigm that counting solutions is at least as hard as deciding existence.
Beyond ordinary set graphs, the authors consider Aczel’s hyper‑extensional digraphs, a generalisation motivated by hypersets (non‑well‑founded set theory). They prove that the recognition problem for hyper‑extensional digraphs, the slim variant, and the associated counting problems all share the same complexity classifications (NP‑complete and #P‑complete respectively). This establishes a tight connection between finite‑set representations and their non‑well‑founded extensions from a complexity viewpoint.
Finally, the paper studies a digraph coding problem inspired by separating codes. An “open‑out‑separating code” is a set of vertices C such that for any two distinct vertices u and v, the intersections of their out‑neighbourhoods with C are different. The authors prove that, given an extensional acyclic digraph and an integer k, deciding whether there exists an open‑out‑separating code of size k is NP‑complete. The reduction again uses a Hamiltonian‑path construction, encoding the need to separate vertices via the code.
Methodologically, all hardness proofs rely on carefully crafted reductions from Hamiltonian Path variants. The reductions embed the path‑finding requirement into the need for distinct out‑neighbourhoods, while simultaneously enforcing acyclicity and, where required, the slim property. For the bipartite‑with‑two‑leaves restriction, the construction introduces a “spine” that forces the two leaves to act as the path endpoints, ensuring that any feasible orientation corresponds to a Hamiltonian path in the original graph.
The results have several implications. First, they settle the long‑standing open question of the algorithmic tractability of set‑graph recognition, showing that no polynomial‑time algorithm is likely unless P=NP, even for highly constrained graph families. Second, the #P‑completeness of the counting variants suggests that exact enumeration of set‑graph representations is infeasible in general, motivating the development of approximation or parameterised algorithms. Third, the extension to hyper‑extensional digraphs bridges finite‑set theory with non‑well‑founded set theory, indicating that the added expressive power does not simplify the underlying computational problems. Finally, the NP‑completeness of the open‑out‑separating code problem links set‑graph theory with coding theory and network diagnostics, opening avenues for interdisciplinary research.
In summary, the paper provides a comprehensive complexity landscape for set graphs, their slim and hyper‑extensional extensions, and associated coding problems. By establishing NP‑completeness and #P‑completeness across a spectrum of natural variants, it delineates the boundaries of tractability and sets the stage for future work on special‑case algorithms, approximation schemes, and applications in areas such as hyperset modelling and graph‑based coding.