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In graph modification problems, we want to modify a graph through some operation (e.g. vertex/edge deletion/addition/contraction) to obtain a graph belonging to some specific class of graphs. Here, we pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. {\it IPEC}, 2024].

That is, we are interested in the parameter $I_e(G)$, which, for a given graph $G$, denotes the smallest $k \geq 0$ such that $G$ can be made locally irregular (\textit{i.e.}, with no two adjacent vertices having the same degree) by deleting $k$ edges. We exhibit notable properties of interest of the parameter ${\rm I}_e$, in general and for particular classes of graphs. Our investigation leads us to propose a conjecture that $I_e(G)$ should always be at most $\frac{1}{3}m+c$, where $m$ is the size of the graph $G$ and $c$ is some constant. We verify this conjecture in the case of paths, cycles, trees and complete graphs, and we provide the first step towards proving the conjecture for cubic graphs. Finally, we present a linear-time algorithm that computes $I_e(G)$ when $G$ is a tree of bounded maximum degree.

[Clara Marcille] (LaBRI)
https://www.labri.fr/perso/pmarcille/

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Remarks / Remarques

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LaBRI/178