List homomorphisms by deleting edges and vertices: tight complexity bounds for bounded-treewidth graphs

TL;DR

The paper investigates LHomVD(H) and LHomED(H), two deletion-variant list-homomorphism problems on bounded-treewidth graphs, and proves tight dichotomies: polynomial-time solvable vs NP-hard depending on fixed H. For vertex deletion, the optimal bounded-treewidth running time is (i(H)+1)^t·n^{O(1)} when H contains certain obstructions, with a matching SETH-based lower bound; for edge deletion, the optimal base is i^⋅(H) and a decomposition-based approach yields improved runtimes in undecomposable cases, also matched by SETH-based lower bounds. The authors introduce a rich gadget toolkit (splitters, translators, matchers, prohibitors) and leverage geometric bi-arc representations to realize complex constraints, enabling both hardness reductions and efficient algorithms. They further strengthen known lower bounds by parameterizing by hub size (σ,δ-hub) and show that the hardness persists even under hub-bounded instances, linking complexity to both treewidth and hub structure. The work unifies and extends classical problems such as Vertex Cover, Max Cut, and Multiway Cut under a single graph-theoretic framework, providing tight, instance-sensitive bounds and a clear path for future extensions to non-list Hom problems.

Abstract

The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. For a fixed $H$, the input of the optimization problem LHomVD($H$) is a graph $G$ with lists $L(v)$, and the task is to find a set $X$ of vertices having minimum size such that $(G-X,L)$ has a list homomorphism to $H$. We define analogously the edge-deletion variant LHomED($H$). This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs $H$ that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed $H$. Second, as our main result, we determine for every graph $H$ for which the problem is NP-hard, the smallest possible constant $c_H$ such that the problem can be solved in time $c^t_H\cdot n^{O(1)}$ if a tree decomposition of $G$ having width $t$ is given in the input.Let $i(H)$ be the maximum size of a set of vertices in $H$ that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD($H$), we show that the smallest possible constant is $i(H)+1$ for every $H$. The situation is more complex for the edge-deletion version. For every $H$, one can solve LHomED($H$) in time $i(H)^t\cdot n^{O(1)}$ if a tree decomposition of width $t$ is given. However, the existence of a specific type of decomposition of $H$ shows that there are graphs $H$ where LHomED($H$) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than $i(H)$. Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed $H$.

Figures (11)

• Figure 1: An example of a graph $H$ that has a decomposition $(A,B,C)$, with $i(H)=k$ and $i^\bullet(H)=3$. The $a_i$'s form an irreflexive independent set and the $b_i$'s form a reflexive clique. Every vertex $a_i$ is adjacent to $\{b_{i},\dots, b_{i+k-1}\}$, and $\{u_1,u_2,u_3\}$ is fully adjacent to every $a_i$ and $b_i$. Observe that $i(H)\geqslant i(H[A])\geqslant k$, as vertices $a_1$, $\dots$, $a_k$ have incomparable neighborhoods. There is no irreflexive edge in $H[A]$, and it can be checked that there is no 3-element set with private or co-private neighbors, implying that $\textsc{LHomED}(H[A])$ is polynomial-time solvable. But $\{u_1,u_2,u_3\}$ has private neighbors, making $\textsc{LHomED}(H)$NP-hard.
• Figure 2: Construction of the $(v,S)$-prohibitor gadget. A dashed line means there is no edge between the two endpoints.
• Figure 3: Representing the interaction of two vertices $u$ and $v$ with $L(v)=\{a_1,\dots,a_5\}$ and $L(u)=\{b_1,\dots, b_4\}$. Black areas denote ones in the interaction matrix.
• Figure 4: An alternating path certifying that $a$ is moved to $B$. Vertex $\ell_4$ is maximal.
• Figure 5: High-level construction of the gadgets in \ref{['lem:vd-gadget']}. Ovals denote the portals of the gadgets with their lists given inside. Solid lines show which transitions between colors of portals are always possible, and dashed lines show transitions that might or might not be possible.

Theorems & Definitions (115)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.3: Decomposition
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: EsmerFMR24hub
• Theorem 1.8: EsmerFMR24hub
• Theorem 2.1