图的一种互可见性着色问题的变体
A variety of the mutual-visibility coloring problem for graphs
- Cochin University of Science and Technology（科钦科技大学）
- University of Maribor（马里博尔大学）
- Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）
- Universidad de Cádiz（加的斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究图互可见性着色的变体，引入对偶、外部和整体互可见性色数，证明判定问题NP完全，并在块图、汉明图和强网格图上给出精确公式或刻画。
AI中文摘要：
本文探讨了基于图可见性性质的顶点着色问题的变体。文章引入并研究了对偶、外部和整体互可见性色数，这些色数将图的顶点集划分为保持特定互可见性条件（称为对偶、外部或整体）的色类。工作提供了这些色参数为有限或无限的结构条件，并确定了判定一个图是否允许使用给定数量的颜色进行对偶、外部或整体互可见性着色是NP完全的，即使限制为两种颜色也是如此。针对几个基本图类，建立了这些色参数的精确公式和紧界。对于块图，基于割顶点和特定禁止子图结构等结构不变量，给出了外部和对偶互可见性色数的完整刻画。在汉明图上，对偶和整体互可见性色数被证明等于因子中较小的维度，而外部互可见性色数被证明等于对应完全二分图的星形荫度。最后，本文研究了强网格图，确定了其整体、外部和对偶互可见性色数的精确值。这些结果展示了参数行为如何根据网格维度从有限常数变化到无穷大。
英文摘要：
This paper explores variations of vertex-coloring problems defined on graph visibility properties. It introduces and studies the dual, outer, and total mutual-visibility chromatic numbers, which partition the vertex set of a graph into color classes that preserve specific mutual-visibility conditions called dual, outer or total. The work provides structural conditions under which these chromatic parameters are finite or infinite, and establishes that deciding whether a graph admits a dual, outer, or total mutual-visibility coloring using a given number of colors is NP-complete, even when restricted to two colors. Exact formulas and tight bounds for these chromatic parameters are established across several fundamental graph classes. For block graphs, complete characterizations are provided for the outer and dual mutual-visibility chromatic numbers based on structural invariants such as cut vertices and specific forbidden subgraph structures. On Hamming graphs, the dual and total mutual-visibility chromatic numbers are shown to equal the smaller dimension of the factors, while the outer mutual-visibility chromatic number is proven to equal the star arboricity of a corresponding complete bipartite graph. Finally, the paper examines strong grid graphs, determining exact values for their total, outer, and dual mutual-visibility chromatic numbers. These results demonstrate how the parameter behaviors range from finite constants to infinity depending on the grid dimensions.
