Graph Irregularity via Edge Deletions

TL;DR

This paper investigates the parameter Ie(G), representing the minimum edges to delete to make a graph locally irregular, providing properties, algorithms, and conjectures, especially focusing on dense graphs and specific graph classes.

Contribution

It introduces parameterized algorithms for computing Ie(G), explores its behavior in dense graphs, and proposes a conjecture relating Ie(G) to the number of edges.

Findings

Two fixed-parameter tractable algorithms for Ie(G)

Exact values of Ie(G) for certain complete graphs

A conjecture that Ie(G) ≤ m/3 + c, verified for various graph families

Abstract

We pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. IPEC, 2024]. That is, we are interested in the parameter Ie(G), which, for a given graph G, denotes the smallest k >= 0 such that G can be made locally irregular (i.e., with no two adjacent vertices having the same degree) by deleting k edges. We exhibit notable properties of interest of the parameter Ie, in general and for particular classes of graphs, together with parameterized algorithms for several natural graph parameters. Despite the computational hardness previously exhibited by this problem (NP-hard, W[1]-hard w.r.t. feedback vertex number, W[1]-hard w.r.t. solution size), we present two FPT algorithms, the first w.r.t. the solution size plus Delta and the second w.r.t. the vertex cover number of the input graph. Finally, we take important steps towards better understanding the behaviour of this problem in dense graphs. This is crucial when considering some of the parameters whose behaviour is still uncharted in regards to this problem (e.g., neighbourhood diversity, distance to clique). In particular, we identify a subfamily of complete graphs for which we are able to provide the exact value of Ie(G). These investigations lead us to propose a conjecture that Ie(G) should always be at most m/3 + c, where $m$ is the number of edges of the graph $G$ and $c$ is some constant. This conjecture is verified for various families of graphs, including trees.