Set graphs. II. Complexity of set graph recognition and similar problems

TL;DR

This paper investigates the computational complexity of recognizing set graphs and related structures, proving NP-completeness and #P-completeness results for various recognition and counting problems, with implications for hereditarily finite sets.

Contribution

It establishes the NP-completeness of set graph recognition, even under restrictive conditions, and extends complexity results to hyper-extensional digraphs and open-out-separating codes.

Findings

Set graph recognition is NP-complete, even for bipartite graphs with two leaves.

Counting set graphs is #P-complete.

Recognition of open-out-separating codes is NP-complete.

Abstract

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.