A Finer View of the Parameterized Landscape of Labeled Graph Contractions

TL;DR

We study the Labeled Contractibility problem, where uniquely labeled graphs $G$ and $H$ are given and the question is whether $H$ can be obtained from $G$ via edge contractions. Building on prior work that is W[1]-hard for the contraction budget and already shows FPT behavior with treewidth via Courcelle’s theorem, we present a constructive dynamic-programming algorithm parameterized by treewidth that runs in $2^{\mathcal{O}(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}$ and prove a matching ETH-based lower bound of $2^{o(\tw^2)}$. The paper also strengthens hardness results by showing NP-hardness under bounded maximum degree and explores parameterizations by solution size plus degeneracy, delivering an improved $ (\delta(H)+1)^k \cdot |V(G)|^{\mathcal{O}(1)}$-time algorithm and a subexponential-impossibility result under ETH. Finally, a brute-force approach is shown to be optimal under ETH, and the authors discuss implications for the broader Maximum Common Labeled Contraction problem. Overall, the work sharpens the landscape for Labeled Contractibility across treewidth, degree, and degeneracy, delivering both practical algorithms and tight complexity bounds.

Abstract

We study the \textsc{Labeled Contractibility} problem, where the input consists of two vertex-labeled graphs $G$ and $H$, and the goal is to determine whether $H$ can be obtained from $G$ via a sequence of edge contractions. Lafond and Marchand~[WADS 2025] initiated the parameterized complexity study of this problem, showing it to be \(\W[1]\)-hard when parameterized by the number \(k\) of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width \(\tw\) of \(G\), via an application of Courcelle's theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for \textsc{Labeled Contractibility} with running time \(2^{\mathcal{O}(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\). We also prove that unless the Exponential Time Hypothesis (Ð) fails, it does not admit an algorithm running in time \(2^{o(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\). This result adds \textsc{Labeled Contractibility} to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains \NP-complete even when both input graphs have bounded maximum degree. We also investigate parameterizations by \((k + δ(G))\) where \(δ(G)\) denotes the degeneracy of \(G\), and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand~[WADS 2025]. We additionally provide an improved \FPT\ algorithm with better dependence on \((k + δ(G))\) than previously known. Finally, we analyze a brute-force algorithm for \textsc{Labeled Contractibility} with running time \(|V(H)|^{\mathcal{O}(|V(G)|)}\), and show that this running time is optimal under Ð.

Figures (5)

• Figure 1: Depicting definitions relevant to the algorithm
• Figure 2: Illustration of the graphs $G'$ and $H'$ in the reduction for the lower bound of Theorem \ref{['thm:treewith']}
• Figure 3: Example of the construction of $G$ with $m=3$. We only show one $x_i$ for convenience that appears in $C_3$
• Figure 4: Illustration of the reduction. (Top) Variable gadgets in $G$ and $H$ for a variable $(v_i)$. (Bottom) Clause gadgets in $G$ and $H$ for some clause of the form $(v_i\lor v_j\lor v_k')$.
• Figure 5: Illustration of the graphs $G$ and $H$ in the reduction. The required adjacency across $\{x_1, x_2\}$ and $A$ forces a candidate solution to form a matching across $A$ and $B$.

Theorems & Definitions (58)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 2.1
• Definition 2.2
• Theorem 5
• Definition 3.1: cygan2015parameterized
• Definition 3.2: cygan2015parameterized