Bounds on Functionality and Symmetric Difference – Two Intriguing Graph Parameters

by   Pavel Dvořák, et al.

[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 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(√( n log n)) and we give a nearly matching Ω(√(n))-lower bound provided by projective planes. Further, we study a related graph parameter symmetric difference, the minimum of |N(u) Δ N(v)| over all pairs of 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(√(n)) ≤sd(INT_n) ≤ O(√(n)). For the related class CA we show that sd(CA_n) = Θ(√(n)). We propose a follow-up question: is fun(CA) bounded?


page 1

page 2

page 3

page 4


Functionality of box intersection graphs

Functionality is a graph complexity measure that extends a variety of pa...

Phase Transitions of Structured Codes of Graphs

We consider the symmetric difference of two graphs on the same vertex se...

Lions and contamination, triangular grids, and Cheeger constants

Suppose each vertex of a graph is originally occupied by contamination, ...

Reconstruction of Line-Embeddings of Graphons

Consider a random graph process with n vertices corresponding to points ...

FASHION: Functional and Attack graph Secured HybrId Optimization of virtualized Networks

Maintaining a resilient computer network is a delicate task with conflic...

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-width

The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedb...

Dominated Minimal Separators are Tame (Nearly All Others are Feral)

A class F of graphs is called tame if there exists a constant k so that ...

Please sign up or login with your details

Forgot password? Click here to reset