[Paper Review] Graph Homomorphism Reconfiguration and Frozen $H$-Colourings

Richard C. Brewster, Jae-baek Lee+3 more|arXiv (Cornell University)|Dec 1, 2017

Advanced Graph Theory Research23 references4 citations

TL;DR

This paper establishes that the reconfiguration problem for $H$-colourings is PSPACE-complete when $H$ is an odd wheel, and provides a complete dichotomy for the existence of frozen $H$-colourings—proving that a graph $G$ admits a frozen $H$-colouring if and only if $H$ contains a non-thermal component that is not a component of $H'$. The results rely on a reduction from a CSP problem shown to be NP-complete via non-existence of a certain polymorphism.

ABSTRACT

For a fixed graph $H$, the reconfiguration problem for $H$-colourings (i.e. homomorphisms to $H$) asks: given a graph $G$ and two $H$-colourings $φ$ and $ψ$ of $G$, does there exist a sequence $f_0,\dots,f_m$ of $H$-colourings such that $f_0=φ$, $f_m=ψ$ and $f_i(u)f_{i+1}(v)\in E(H)$ for every $0\leq i

Motivation & Objective

• To determine the computational complexity of the $H$-reconfiguration problem for homomorphisms to a fixed graph $H$, particularly when $H$ is an odd wheel.
• To characterize the existence of frozen $H$-colourings—$H$-colourings that cannot be reconfigured to any other distinct $H$-colouring.
• To establish a complete dichotomy for the existence of frozen $H$-colourings in terms of structural properties of $H$.
• To explore connections between reconfiguration complexity and the existence of polymorphisms in constraint satisfaction problems.
• To investigate the diameter of components in the reconfiguration graph $\mathbf{Hom}(G,H)$, with implications for NP-completeness of the reconfiguration problem.

Proposed method

• Reduction from a constraint satisfaction problem (CSP) shown to be NP-complete via the non-existence of a specific polymorphism.
• Construction of auxiliary graphs $H'$ and $J$ to model reconfiguration dynamics, particularly focusing on components with non-bipartite structure.
• Use of projection maps and local surjectivity to analyze the image of neighbourhoods under $H$-colourings.
• Application of Lemma 3.7 to show that a colouring is frozen if the image of every vertex's neighbourhood lies in a distinguishing set $\mathcal{D}$.
• Leveraging Corollary 3.4 and Lemma 3.21(b) to ensure that components of $H'$ and their lifts in $\tilde{J}$ preserve structural constraints.
• Proof by contradiction to rule out the existence of frozen colourings when $H$ contains a $K_2$ component not in $H'$, exploiting the non-bipartite nature of components in $\tilde{J}$.

Experimental results

Research questions

• RQ1 Is the $H$-reconfiguration problem PSPACE-complete for $H$ being an odd wheel?
• RQ2 For which graphs $H$ does there exist a graph $G$ that admits a frozen $H$-colouring?
• RQ3 What structural properties of $H$ determine whether a frozen $H$-colouring can exist in some $G$?
• RQ4 Can the existence of a polymorphism be used to characterize tractability in $H$-reconfiguration?
• RQ5 Does the diameter of components in $\mathbf{Hom}(G,H)$ grow polynomially in $|V(G)|$ for all $H$?

Key findings

• The $H$-reconfiguration problem is PSPACE-complete when $H$ is an odd wheel.
• A graph $G$ admits a frozen $H$-colouring if and only if $H$ contains a non-thermal component that is not a component of $H'$, where $H'$ is the graph formed by removing all thermal components from $H$.
• The existence of a frozen $H$-colouring is decidable in polynomial time via a structural characterization of $H$.
• The proof of NP-completeness for the underlying CSP relies on the non-existence of a certain polymorphism, establishing hardness through universal algebraic techniques.
• Components of $\tilde{J}$ are non-bipartite, which rules out the possibility of certain $H$-colourings when $H$ contains a $K_2$ component not in $H'$, thus excluding such $H$ from admitting frozen colourings.
• The reconfiguration graph $\mathbf{Hom}(G,H)$ may have superpolynomial diameter for some $H$, as shown by prior results for $K_k$ with $k \geq 4$, suggesting that $H$-reconfiguration is not in NP unless component diameters are bounded polynomially.