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.

翻译：受1-2-3猜想里程碑式解决的启发，我们首次研究了顶点染色{0,1}-边赋权问题及其推广形式——顶点染色预边赋权问题在多种结构参数下的参数化复杂性。基础问题——顶点染色{0,1}-边赋权要求判断能否为图的每条边从{0,1}中分配一个权重，使得每对相邻顶点各自的关联边权重之和互异。在顶点染色预边赋权变体中，我们给定一张图，其中部分边已被赋予{0,1}中的固定权重，目标是判断该部分赋权能否扩展至所有剩余边，使得最终完整赋值满足顶点染色性质。尽管此类赋权在特定图类中的存在性已被充分理解，但其在结构参数化下的算法复杂性此前尚未被探索。我们针对一系列结构参数层次，证明了该问题的难解性与可解性：当以输入图的反馈顶点集大小为参数时，基础问题与预边赋权变体均为W[1]-难解；在积极方面，我们证明基础问题及预赋权重均为1的受限预边赋权变体，在输入图的顶点覆盖大小为参数时属于FPT；此外，我们还表明基础问题与预边赋权变体在输入图的树宽为参数时均存在XP算法。