AI 中文总结
研究单变量依赖图类的邻域复杂度和半径为1的合并宽度，证明该类有几乎线性邻域复杂度及半径为1的合并宽度\(n^{o(1)}\)，给出\(\mathcal{O}(n^5)\)时间算法计算见证特定合并宽度的构造序列，解决相关猜想并提供结构描述。
AI 中文摘要
单变量依赖是关于遗传图类上一阶模型检查固定参数可处理性的一条结构分界线。若所有图的类不能用固定一阶公式在其顶点着色成员中解释，则图类是单变量依赖的。我们证明了单变量依赖的两个结构结果。首先，每个单变量依赖类具有几乎线性邻域复杂度：对于该类中的每个图\(G\)和每个集合\(A\subseteq V(G)\)，族\(\{N_G(v)\cap A: v\in V(G)\}\)的大小为\(|A|^{1 + o(1)}\)。其次，单变量依赖类中的每个\(n\)顶点图具有半径为1的合并宽度\(n^{o(1)}\)。这解决了单变量依赖与几乎有界合并宽度之间猜想联系的半径为1的情况，并提供了单变量依赖图类基于分解的首个结构描述。我们的证明是算法性的：给出一个\(\mathcal{O}(n^5)\)时间算法，给定一个\(n\)顶点图\(G\)，对于每个\(A\subseteq V(G)\)有\(|\{N_G(v)\cap A: v\in V(G)\}|\leq O(|A|^d)\)，计算一个见证半径为1的合并宽度为\(\mathcal{O}(n^{1 - 1/d}\log n)\)的构造序列。
英文摘要
Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is \emph{monadically dependent} if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has \emph{almost linear neighborhood complexity}: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toruńczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.
