Bounds on Functionality and Symmetric Difference – Two Intriguing Graph Parameters

[Alecu et al.: Graph functionality, JCTB2021] define functionality (fun), a graph parameter that generalizes graph degeneracy. They research the relation of functionality to many other graph parameters (tree-width, clique-width, VC-dimension, etc.). Extending their research, we prove logarithmic lower bound for functionality of random graph G(n,p) for large range of p. Previously known graphs have functionality logarithmic in number of vertices. We show that for every graph G on n vertices we have fun(G) ≤ O(sqrt(n log n)) and we give a nearly matching Ω(sqrt(n))-lower bound provided by projective planes. Further, we study a related graph parameter symmetric difference (sd), the minimum size of the symmetric difference between the neighbourhood of any two distinct vertices of the “worst possible” induced subgraph. It was observed by Alecu et al. that fun(G) ≤ sd(G)+1 for every graph G. We compare fun and sd for the class INT of interval graphs and CA of circular-arc graphs. We let INT_n denote the n-vertex interval graphs, similarly for CA_n. Alecu et al. ask, whether fun(INT) is bounded. Dallard et al. answer this positively in a recent preprint. On the other hand, we show that Ω(4sqrt(n)) ≤ sd(INT_n) ≤ O(3sqrt(n)). For the related class CA we show that sd(CA_n) = Θ(sqrt(n)). We propose a follow-up question: is fun(CA) bounded?