Sarosh Adenwalla, Samuel Braunfeld, John Sylvester, Viktor Zamaraev
原文摘要
Boolean combinations allow combining given combinatorial objects to obtain new, potentially more complicated, objects. In this paper, we initiate a systematic study of this idea applied to graphs. In order to understand expressive power and limitations of boolean combinations in this context, we investigate how they affect different combinatorial and structural properties of graphs, in particular $χ$-boundedness, as well as characterize the structure of boolean combinations of graphs from various classes.
John Sylvester, Viktor Zamaraev, Maksim Zhukovskii
原文摘要
A hereditary graph class is small if it contains at most $n!c^n$ labelled graphs on $n$ vertices for some constant $c$. Small classes include proper minor-closed classes and classes of bounded twin-width. We prove that every small class has linear neighbourhood complexity, that is, in every graph in the class, vertex neighbourhoods intersect any $k$-vertex set in only $O(k)$ distinct ways. This resolves a conjecture of Bonnet, Duron, and the first two authors (ITCS 2025) and makes the algorithmic, structural, and representation consequences of linear neighbourhood complexity available for every small class. In particular, every small class admits an adjacency labelling scheme with $O(\log^2 n)$-bit labels, improving the previous $O(\log^3 n)$ bound.