发表机构
University of Regina; Alberta Machine Intelligence Institute（里贾纳大学; 阿尔伯塔机器智能研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出符号图上的双色燃烧模型，通过汉明距离刻画多元数，给出符号路径的精确值，并证明相关计算问题的困难性。
AI 中文摘要
我们引入并分析了一种新的图燃烧模型，其中两场竞争的火（分别染成黄色和绿色）点燃给定符号图的顶点并沿其边传播。边的布尔符号决定火沿边传播时是否改变颜色。在每一步中，玩家以自己选择的颜色点燃一个新顶点，而先前激活的火继续传播。给定一个符号图 $\Gamma$，目标是燃烧尽可能多的单一颜色的顶点；可达到的最优值称为 $\Gamma$ 的多元数。我们通过到与 $\Gamma$ 关联的二元码的汉明距离来表达多元数。特别地，在 $\Gamma$ 的切换类上，多元数的最小值等于顶点数减去该码的覆盖半径。在签名的某些条件下，我们确定了符号路径的多元数的精确值。相比之下，我们证明了确定或近似多元数的多个变体问题的困难性结果。
英文摘要
We introduce and analyze a new model of graph burning, in which two competing fires (coloured yellow and green) ignite vertices of a given signed graph and propagate along its edges. The boolean sign of an edge determines whether a fire spreading along the edge changes colour or not. In each step, a player ignites a new vertex in a colour of their choice, while previously activated fires continue to spread. Given a signed graph $Γ$, the objective is to burn the maximum possible number of vertices in a single colour; the optimal achievable value is called the plurality number of $Γ$. We express the plurality number through Hamming distances to a binary code associated with $Γ$. In particular, the minimum plurality number over the switching class of $Γ$ equals the number of vertices minus the covering radius of this code. Under certain conditions on the signature, we determine exact values of the plurality number of signed paths. By contrast, we prove hardness results for multiple variants of the problem of determining or approximating the plurality number.
