三角图的鲁棒代数理论
Robust Algebraic Theories of Triangle Graphs
浏览论文内容
中文总结 AI 辅助
该研究对三角图及扇图给出代数刻画，扩展系列并行图代数至树宽3，证明CMSO在其上下文无关集合上可判定，且可识别图语言与CMSO可定义语言一致。
中文摘要 AI 辅助
三角图是树宽至多为3且每条边都属于一个三角形的图，该类包含阿波罗尼亚网络等知名图族。我们还研究扇图，它是三角图的子类，与3-连通三角图密切相关。主要结果是对这两类图给出代数刻画：我们引入基于并行复合和三元序列复合的两种图代数，证明它们分别恰好生成三角图和扇图。这些代数将经典的系列并行图代数从树宽2自然扩展至树宽3。基于上述刻画，我们研究上下文无关、可识别及逻辑可定义的图语言，证明计数 monadic 二阶逻辑（CMSO）在三角图和扇图的上下文无关集合上可判定，且对这两种代数，可识别图语言与CMSO可定义语言一致。
英文摘要
Triangle graphs are graphs of tree-width at most three in which every edge belongs to a triangle. This class encompasses well-known graph families such as Apollonian networks. We also consider fan graphs, a subclass of triangle graphs closely related to the 3-connected triangle graphs. Our main result is an algebraic characterization of both classes. We introduce two graph algebras based on parallel composition and a ternary serial composition, and show that they generate exactly the triangle and fan graphs, respectively. These algebras provide a natural extension of the classical algebra of series-parallel graphs from tree-width two to tree-width three. Building on these characterizations, we investigate context-free, recognizable, and logically-definable graph languages. We show that counting monadic second-order logic (CMSO) is decidable over the context-free sets of triangle and fan graphs. Moreover, we prove that recognizable graph languages coincide with languages definable in CMSO for both algebras.
