On Structural Parameterizations of Load Coloring
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Given a graph G and a positive integer k, the 2-Load coloring problem is to check whether there is a 2-coloring f:V(G) →{r,b} of G such that for every i ∈{r,b}, there are at least k edges with both end vertices colored i. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (Algorithmica 79:211-229, 2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result. In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that cannot be solved in time f(w)n^o(w), unless ETH fails and it can be solved in time n^O(w), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that is NP-complete even on bipartite graphs and split graphs.
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