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arXiv · 2609.27719

Vertex-Coloring Edge-Weighting: Kernelization and Generalization

Shubhada Aute·Fahad Panolan·Geevarghese Philip
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Abstract

An edge weighting of a graph induces a coloring of its vertices in which the color of a vertex is the total weight of the edges incident with it. Such an edge weighting is proper if adjacent vertices always receive distinct colors. Deciding whether a graph admits a proper weighting is known to be NP-complete for the weight set $\{0,1\}$, and also for $\{1,2\}$. In recent work (arXiv:2604.12363) we showed that both problems are FPT parameterized by the vertex cover number $k$, but it was open -- to the best of our knowledge -- whether either parameterized problem had a polynomial kernel. In this work, we show that both problems have polynomial kernels when parameterized by $k$. We also show that both problems are W[1]-hard parameterized by treedepth, answering another question from our earlier work. We then study the pre-weighted versions of the two problems, in which the weights of some edges are fixed in advance, and the task is to extend the assignment to a proper weighting of the whole graph. We show that both pre-weighted problems are FPT parameterized by the vertex cover number $k$. For the $\{1,2\}$ version the running time is $2^{O(k \log k)} \cdot n$; for the $\{0,1\}$ version we obtain the same running time when every pre-weight is $1$, and a slower FPT algorithm in the general case. We also show that both pre-weighted problems are W[1]-hard parameterized by either of (i) the feedback vertex set number or (ii) the treedepth of the input graph. Since a graph with no pre-assigned weights is a special case, our algorithms for the pre-weighted versions solve the two original problems as well, in time $2^{O(k \log k)} \cdot n$, significantly improving on the bound of $2^{O(k^4)} \cdot n^{O(1)}$ from our earlier work.

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Shubhada Aute, Fahad Panolan, Geevarghese Philip. 2026-09-23. Vertex-Coloring Edge-Weighting: Kernelization and Generalization. https://arxiv.org/abs/2609.27719

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