Incremental List Coloring of Graphs, Parameterized by Conservation

Abstract

Incrementally k-list coloring a graph means that a graph is given by adding stepwise one vertex after another, and for each intermediate step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing some of in total k colors. We introduce the “conservative version” of this problem by adding a further parameter c ∈ ℕ specifying the maximum number of vertices to be recolored between two subsequent graphs (differing by one vertex). This “conservation parameter” c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing radical changes. We show that the problem is NP-hard for k ≥ 3 and W[1]-hard when parameterized by c. In contrast, the problem becomes fixed-parameter tractable with respect to the combined parameter (k,c). We prove that the problem has an exponential-size kernel with respect to (k,c) and there is no polynomial-size kernel unless NP ⊆ coNP/poly. Finally, we provide empirical findings for the practical relevance of our approach in terms of an effective graph coloring heuristic.