Abstract
[Alecu et al.: Graph functionality, JCTB2021] define functionality, 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 a 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 \(\textrm{fun}(G) \le O(\sqrt{ n \log n})\) and we give a nearly matching \(\varOmega (\sqrt{n})\)-lower bound provided by projective planes.
Further, we study a related graph parameter symmetric difference, the minimum of \(|N(u) \mathrm {\Delta }N(v)|\) over all pairs of vertices of the “worst possible” induced subgraph. It was observed by Alecu et al. that \(\textrm{fun}(G) \le \textrm{sd}(G)+1\) for every graph G. We compare \(\textrm{fun}\) and \(\textrm{sd}\) for the class \(\textrm{INT}\) of interval graphs and \(\textrm{CA}\) of circular-arc graphs. We let \(\textrm{INT}_n\) denote the n-vertex interval graphs, similarly for \(\textrm{CA}_n\).
Alecu et al. ask, whether \(\textrm{fun}(\textrm{INT})\) is bounded. Dallard et al. answer this positively in a recent preprint. On the other hand, we show that \(\varOmega (\root 4 \of {n}) \le \textrm{sd}(\textrm{INT}_n) \le O(\root 3 \of {n})\). For the related class \(\textrm{CA}\) we show that \(\textrm{sd}(\textrm{CA}_n) = \varTheta (\sqrt{n})\).
We propose a follow-up question: is \(\textrm{fun}(\textrm{CA})\) bounded?
P. Dvořák—Supported by Czech Science Foundation GAČR grant #22-14872O.
L. Folwarczný—Supported by Czech Science Foundation GAČR grant 19-27871X.
M. Opler—Supported by Czech Science Foundation GAČR grant 22-19557S.
P. Pudlák—Supported by Czech Science Foundation GAČR grant 19-27871X.
R. Šámal—Partially supported by grant 22-17398S of the Czech Science Foundation. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115).
T. A. Vu—Supported by Czech Science Foundation GAČR grant 22-22997S.
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Acknowledgements
The research presented in this paper has been started during the KAMAK workshop in 2021. We are grateful to the organizers of this wonderful event.
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Dvořák, P., Folwarczný, L., Opler, M., Pudlák, P., Šámal, R., Vu, T.A. (2023). Bounds on Functionality and Symmetric Difference – Two Intriguing Graph Parameters. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_22
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