Graph Irregularity via Edge Deletions

TL;DR

We study edge-irregulators that render a graph locally irregular by deleting edges, formalizing $I_e(G)$ as the minimum size of an edge-set to remove and defining the Irregularity-Deletion decision problem. The paper proves NP-hardness and W[1]-hardness for natural parameters, and then provides two FPT algorithms: one parameterized by the solution size $k$ plus the maximum degree $\Delta$, and another parameterized by the vertex cover number. It derives a kernel of size $O(k \Delta^{2k+2})$ and shows membership in FPT for clique-width plus $\Delta$ via an $MSO_2$ formulation, while computing exact values for several families (paths, cycles, $K_{n,m}$) and analyzing triangular-order complete graphs to obtain $I_e(K_{t_k})$, all supporting the conjecture $I_e(G) \le \frac{|E(G)|}{3}+c$. The results advance understanding of $I_e$ in dense graphs and trees and outline future directions for tightening bounds and extending to additional graph parameters.

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.