AI 中文总结
研究有界树宽和路径宽图的邻域复杂度，给出\(k\geq w + 1\)时树宽和路径宽图邻域复杂度的上界及达到这些界的构造，还得出路径宽或树宽为\(1\)的图的邻域复杂度紧界。
AI 中文摘要
图\(G\)的邻域复杂度\(nc(G,k)\)是衡量对于图\(G\)和整数\(k\)，在所有大小为\(k\)的顶点子集\(S\)上，\(G\)中\(S\) - 邻域\(|\{N[v]\cap S, v\in V(G)\}|\)的最大可能数量。该概念在结构图论和算法设计中很重要。一般\(nc(G,k)\leq 2^k\)，稀疏图和结构化密集图有线性邻域复杂度。本文聚焦有界树宽和路径宽的图，证明了\(k\geq w + 1\)时的两个不等式，并给出达到这些界的构造。有趣的是，路径宽或树宽为\(1\)的图有\(nc(G,k)\leq\frac{7}{3}k\)。
英文摘要
The neighbourhood complexity $nc(G,k)$ of a graph $G$ is a quantity measuring, for a graph $G$ and an integer $k$, the maximum possible number (over all vertex subsets $S$ of size $k$) $|\{N[v]\cap S, v\in V(G)\}|$ of $S$-neighbourhoods in $G$. This notion is important in structural graph theory and algorithm design (especially in parameterized complexity, in particular model checking and kernelization). While generally $nc(G,k)\leq 2^k$ and this bound can be achieved, it is known that sparse graphs and structured dense graphs have linear neighbourhood complexity, that is, $nc(G,k)\in O(k)$ for any such graph $G$. However, for many graph classes, the best possible constants are not known. We focus on graphs of bounded treewidth and pathwidth, showing that (when $k\geq w+1$) (i) if $G$ has treewidth $w\geq 2$, then $nc(G,k)\leq (k-w+1)2^{w}+w$, and (ii) if $G$ has pathwidth $w\geq 2$, then $nc(G,k)\leq (k-w+2)2^{w-1}+2k-w-2$. Moreover, we provide constructions that reach these bounds, whenever $w\geq 2$ and $k\geq 2w+1$ ($k\geq 2w-1$ for pathwidth). Interestingly, in contrast, we also have the tight bound $nc(G,k)\leq \frac{7}{3}k$, for graphs $G$ with pathwidth 1 or treewidth 1.
