Embeddability of graphs and Weihrauch degrees

TL;DR

The paper advances the systematic classification of the uniform computational content of (induced) subgraph problems for fixed countable graphs using effective Wadge reducibility and Weihrauch reducibility. It reveals a sharp dichotomy: the induced-subgraph decision problem for infinite graphs is always $oldsymbol{ Sigma}^1_1$-complete, while the corresponding search problems exhibit a rich spectrum, with induced-subgraph copies typically Weihrauch-equivalent to $oldsymbol{C}_{ N^ }$ and non-induced subgraph copies ranging from $oldsymbol{C}_{ N^ }$ to lim-type degrees. A novel tool, the finite part of a problem, clarifies how first-order content interacts with higher-type computations, and the analysis of the ray $R$-substructure shows both strong nontrivial separations and surprising compositions like $oldsymbol{C}_{ N^ } otrac{W}{=} ext{S-Copy}_{R}$ yet $oldsymbol{C}_{ N^ } uildrel{}ar{} = ext{lim}_2* ext{S-Copy}_{R}$. The results connect reverse mathematics and computable analysis in graph-theoretic contexts, map graph-structural properties to Weihrauch degrees, and raise open questions about finer gradations within the induced/subgraph landscape and related supergraph variants.

Abstract

We study the complexity of the following related computational tasks concerning a fixed countable graph G: 1. Does a countable graph H provided as input have a(n induced) subgraph isomorphic to G? 2. Given a countable graph H that has a(n induced) subgraph isomorphic to G, find such a subgraph. The framework for our investigations is given by effective Wadge reducibility and by Weihrauch reducibility. Our work follows on "Reverse mathematics and Weihrauch analysis motivated by finite complexity theory" (Computability, 2021) by BeMent, Hirst and Wallace, and we answer several of their open questions.

Figures (4)

• Figure 1: Some of the multi-valued functions studied in this paper. Black arrows represent Weihrauch reducibility in the direction of the arrow. Here $F$ represent a finite graph, $G$ an infinite c.e. graph, $R$ a c.e. graph such that $\mathsf{R}\subseteq_{\mathbf{s}} R$ and $\mathsf{G} \in \{ \mathsf{T}_{2k+1}, \mathsf{F}_{2k+2}\}$.
• Figure 2: A summary of the Weihrauch reductions between problems considered in § \ref{['particularcase']}. As in Figure \ref{['SummaryAtThebeginningGraphs']} black arrows represent Weihrauch reducibility in the direction of the arrow while red arrows represent the absence of a Weihrauch reduction in the direction of the arrow.
• Figure 3: On the left side, the disconnected union $\bigotimes_{i>2}C_{i}$ of all cyclic graphs, shown up to $C_5$. On the right side, the connected union $\bigodot_{i>2}C_{i}$ of all cyclic graphs, shown up to $C_5$: starting from the left one, the two red vertices denote respectively the vertices $\langle \mathsf{w}_3,\mathsf{v}_4 \rangle$ and $\langle \mathsf{w}_4,\mathsf{v}_5 \rangle$.
• Figure 4: Dashed arrows represent Weihrauch reducibility in the direction of the arrow, leaving open whether the reduction is strict.

Theorems & Definitions (125)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Definition 2.5: valentisolda
• Theorem 2.6: dzafarovsolomonyokoyama
• Definition 2.7
• Proposition 2.8
• proof
• Definition 2.9