The Parameterized Complexity of Vertex-Coloring Edge-Weighting

2026-04-14

AI Summary

This study investigates whether edges of a graph can be assigned weights from {0,1} such that the weighted degrees of adjacent vertices are distinct, and further examines the extendability of such weightings when some edges are pre-assigned weights. Employing parameterized complexity theory, the paper systematically analyzes the computational complexity of this problem and its pre-weighted variant under structural parameters including feedback vertex set, vertex cover, and treewidth. The main contributions are threefold: it is the first to apply parameterized complexity to the vertex-distinguishing edge-weighting problem; it establishes W[1]-hardness with respect to the feedback vertex set parameter; it presents an FPT algorithm parameterized by vertex cover; and it provides an XP algorithm for treewidth, thereby delineating the boundary between tractability and intractability for this problem.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Motivated by the landmark resolution of the 1-2-3 Conjecture, we initiate the study of the parameterized complexity of the Vertex-Coloring {0,1}-Edge-Weighting problem and its generalization, Vertex-Coloring Pre-edge-Weighting, under various structural parameters. The base problem, Vertex-Coloring {0,1}-Edge-Weighting, asks whether we can assign a weight from {0,1} to each edge of a graph. The goal is to ensure that for every pair of adjacent vertices, the sums of their incident edge weights are distinct. In the Vertex-Coloring Pre-edge-Weighting variant, we are given a graph where a subset of edges is already assigned fixed weights from {0,1}. The goal is to determine if this partial weighting can be extended to all remaining edges such that the final, complete assignment satisfies the proper vertex coloring property. While the existence of such weightings is well-understood for specific graph classes, their algorithmic complexity under structural parameterization has remained unexplored. We prove both hardness and tractability for the problem, across a hierarchy of structural parameters. We show that both the base problem and the Pre-edge-Weighting variant are W[1]-hard when parameterized by the size of a feedback vertex set of the input graph. On the positive side, we establish that the base problem and a restricted Pre-edge-Weighting variant where the pre-assigned weights are all 1, become FPT when parameterized by the size of a vertex cover of the input graph. Further, we show that both the base problem and the Pre-edge-Weighting variant have XP algorithms when parameterized by the treewidth of the input graph.

Problem

• Vertex-Coloring Edge-Weighting
• Parameterized Complexity
• Structural Parameters
• Pre-edge-Weighting
• Graph Coloring

Innovation

• parameterized complexity
• vertex-coloring edge-weighting
• feedback vertex set
• vertex cover
• treewidth