关于燃烧博弈：Nordhaus-Gaddum 界与图乘积
On the Burning Game: Nordhaus-Gaddum Bounds and Graph Products
- University of Primorska（波雷克大学）
- Faculty of Mathematics and Physics, University Ljubljana（卢布尔雅那大学数学物理学院）
- Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那数学、物理与力学研究所）
- Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）
- University of Rhode Island（罗德岛大学）
- Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院数学与信息系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究图上的燃烧博弈，建立了博弈燃烧数的 Nordhaus-Gaddum 界，并针对强、笛卡尔、字典序和冠状四种图乘积给出了相应的界。
AI中文摘要：
我们继续研究图上的燃烧博弈。给定一个图 $G$，两位玩家——燃烧者（Burner）和拖延者（Staller）——轮流选择 $G$ 中的顶点进行燃烧。所有已燃烧的顶点会像燃烧过程一样，将火蔓延至未燃烧的相邻顶点。燃烧者的目标是尽快烧完整个图，而拖延者则希望过程尽可能持久。若双方均以最优策略进行，当燃烧者先手时，烧完整个图 $G$ 所需的时间步数称为博弈燃烧数 $b_{\ m g}(G)$；若拖延者先手，则称为拖延者先手博弈燃烧数 $b_{\ m g}'(G)$。本文进一步研究该博弈，建立了博弈燃烧数的 Nordhaus-Gaddum 界，并给出了四种不同类型图乘积的界：强乘积、笛卡尔乘积、字典序乘积和冠状乘积。
英文摘要：
We continue research on the burning game on graphs. Given a graph $G$, two players, Burner and Staller, take turns in selecting vertices of $G$ to burn. All burned vertices spread fire to unburned neighboring vertices, as in the burning process. The goal of Burner is to burn the graph as quickly as possible, while Staller wants the process to last as long as possible. If both players play optimally, then the number of time steps needed to burn the whole graph $G$ is the game burning number $b_{\rm g}(G)$ if Burner makes the first move, and the Staller-start game burning number $b_{\rm g}'(G)$ if Staller starts. In this paper, we study this game further, establishing Nordhaus-Gaddum bounds on the game burning number, as well as bounds for four different types of graph products: strong, Cartesian, lexicographic and corona products.
