On Structural Parameterizations of Load Coloring

TL;DR

This paper investigates the computational complexity of the 2-Load coloring problem, showing its fixed-parameter tractability with respect to certain graph parameters and NP-completeness on specific graph classes.

Contribution

It establishes the parameterized complexity of the problem relative to structural graph parameters like clique-width and cluster distances, and proves NP-completeness on bipartite and split graphs.

Findings

FPT algorithm for parameters like distance to cluster, co-cluster, and threshold graphs.

No $f(w)n^{o(w)}$ time algorithm for clique-width unless ETH fails.

NP-completeness on bipartite and split graphs.

Abstract

Given a graph $G$ and a positive integer $k$, the 2-Load coloring problem is to check whether there is a $2$-coloring $f:V(G) \rightarrow \{r,b\}$ of $G$ such that for every $i \in \{r,b\}$, 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 \lcp{} 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 \lcp{} is NP-complete even on bipartite graphs and split graphs.