The Parameterized Complexity of Vertex-Coloring Edge-Weighting
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.