A mixed graph contains (undirected) edges as well as (directed) arcs, thus generalizing undirected and directed graphs. A proper coloring $c$ of a mixed graph $G$ assigns a positive integer to each vertex such that $c(u)\neq c(v)$ for every edge $\{u,v\}$ and $c(u)<c(v)$ for every arc $(u,v)$ of $G$. As in classical coloring, the objective is to minimize the number of colors. Thus, mixed (graph) coloring generalizes classical coloring of undirected graphs and allows for more general applications, such as scheduling with precedence constraints, modeling metabolic pathways, and process management in operating systems; see a survey by Sotskov [Mathematics, 2020]. We initiate the systematic study of the parameterized complexity of mixed coloring. We focus on structural graph parameters that lie between cliquewidth and vertex cover, primarily with respect to the underlying undirected graph. Unlike classical coloring, which is fixed-parameter tractable (FPT) parameterized by treewidth or neighborhood diversity, we show that mixed coloring is W[1]-hard for treewidth and even paraNP-hard for neighborhood diversity. To utilize the directedness of arcs, we introduce and analyze natural generalizations of neighborhood diversity and cliquewidth to mixed graphs, and show that mixed coloring becomes FPT when parameterized by mixed neighborhood diversity. Further, we investigate how these parameters are affected if we add transitive arcs, which do not affect colorings. Finally, we provide tight bounds on the chromatic number of mixed graphs, generalizing known bounds on mixed interval graphs.

翻译：混合图同时包含（无向）边和（有向）弧，从而推广了无向图和有向图。混合图$G$的正常着色$c$为每个顶点赋一个正整数，使得对$G$中每条边$\{u,v\}$有$c(u)\neq c(v)$，并对每条弧$(u,v)$有$c(u)<c(v)$。与经典着色类似，其目标是使所用颜色数最小化。因此，混合（图）着色推广了无向图的经典着色，并可应用于更广泛的场景，如带有优先级约束的调度、代谢通路建模及操作系统的进程管理（参见Sotskov的综述[Mathematics, 2020]）。我们首次系统研究了混合着色的参数化复杂性。我们聚焦于介于团宽和顶点覆盖之间的结构图参数（主要针对底层无向图）。与经典着色（以树宽或邻域多样性为参数时属于固定参数可解问题）不同，我们证明混合着色在树宽参数下为W[1]-难问题，在邻域多样性参数下甚至为paraNP-难问题。为利用弧的有向性，我们引入并分析了邻域多样性和团宽在混合图上的自然推广，并证明当以混合邻域多样性为参数时混合着色是固定参数可解的。此外，我们研究了添加不影响着色的传递弧如何影响这些参数。最后，我们给出混合图色数的紧界，推广了已知的混合区间图的界。