Neighbourhood complexity and identification problems for graphs of bounded treewidth and pathwidth

2026-07-18

AI Summary

This work investigates upper bounds on the neighborhood complexity of graphs with bounded treewidth and pathwidth, where neighborhood complexity measures the number of distinct neighborhoods induced by vertex subsets of a given size. By leveraging graph decompositions, combinatorial counting, and extremal graph theory, the authors establish the first tight linear upper bound for graphs of treewidth $w \geq 2$, namely $\mathrm{nc}(G,k) \leq (k - w + 1)2^w + w$. For graphs of pathwidth $w$, they derive an improved bound $\mathrm{nc}(G,k) \leq (k - w + 2)2^{w-1} + 2k - w - 2$. In the special case $w = 1$, they prove the optimal constant-factor bound $\mathrm{nc}(G,k) \leq 7k/3$. All bounds are shown to be tight via explicitly constructed families of graphs, thereby resolving open questions concerning exact extremal values in this setting.

Abstract

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.

Problem

• neighbourhood complexity
• treewidth
• pathwidth
• graph classes
• upper bounds

Innovation

• neighbourhood complexity
• treewidth
• pathwidth
• tight bounds
• parameterized complexity