对于秩宽至多为二的图，可识别性等同于CMSO可定义性
Recognizability equals CMSO-definability for graphs of rank-width at most two
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中文总结 AI 辅助
研究了秩宽至多为二的有限图，证明VR可识别性与计数一元二阶可定义性一致，推进了可识别性与可定义性问题，通过处理分裂素图等一系列步骤完成证明，并将结果从素图推广到任意此类图。
中文摘要 AI 辅助
我们证明，在秩宽至多为二的有限图上，VR可识别性与计数一元二阶可定义性是一致的。这将可识别性与可定义性问题从有界线性团宽推进到秩宽为一的分裂分解情况之外的第一个非平凡有界秩宽层次。证明首先处理分裂素图。Clark和Whittle的最大部分树理论组织了非顺序割秩为二的分离，而单个强分离确定了所有强等价类，并产生了一个CMSO可定义的规范核分层族。尽管辅助部分树本身不是可转换的，但它证明了每个规范局部片段都有一个线性秩宽一致有界的端口连续布局。宽度论证使用划分原子以及Hall、Oxley、Semple和Whittle的分支宽为三的展示定理，并且不假设图躯干保持素性。相干有序秩为二的框架允许进行有限状态的自底向上评估，其局部转换由Bojańczyk、Grohe和Pilipczuk的有界线性团宽定理定义。最后，CMSO可转换的规范分裂分解将结果从素图提升到秩宽至多为二的任意图。
英文摘要
We prove that, on finite graphs of rank-width at most two, VR-recognizability and counting monadic second-order definability coincide. This advances the recognizability-versus-definability problem from bounded linear clique-width to the first nontrivial bounded rank-width level beyond the rank-width-one split-decomposition case. The proof first treats split-prime graphs. The maximal partial-tree theory of Clark and Whittle organizes the non-sequential cut-rank-two separations, while a single strong separation orients all strong equivalence classes and yields a CMSO-definable laminar family of canonical cores. Although the auxiliary partial tree is not itself transduced, it proves that every canonical local piece has a port-contiguous layout of uniformly bounded linear rank-width. The width argument uses partition atoms and the branch-width-three display theorem of Hall, Oxley, Semple, and Whittle and does not assume that graph torsos remain prime. Coherent ordered rank-two frames then permit a finite-state bottom-up evaluation whose local transitions are definable by the bounded-linear-clique-width theorem of Bojańczyk, Grohe, and Pilipczuk. Finally, the CMSO-transducible canonical split decomposition lifts the result from prime graphs to arbitrary graphs of rank-width at most two.
发表机构
- American University of the Middle East（中东美国大学）
机构由 AI 辅助整理，请以论文原文为准。
