On the hardness of determining the irregularity strength of graphs

Abstract

Let G be a graph, and ℓ:E(G)→{1,…,k} be a k-labelling of G, i.e., an assignment of labels from {1,…,k} to the edges of G. We say that ℓ is irregular if no two distinct vertices of G are incident to the same sum of labels. The irregularity strength of G, denoted by s(G), is the smallest k such that irregular k-labellings of G exist. These notions were introduced in the late 1980s as an alternative way to deal with an optimisation problem where one aims at making a graph irregular by multiplying its edges in an optimal way. Since then, the irregularity strength has received a lot of attention, focusing mainly on proving bounds and investigating side aspects and variants.In this work, we consider the algorithmic complexity of determining the irregularity strength of a given graph. We prove that two close variants of this problem are NP-hard, which we suspect might indicate that the original problem is hard too. Namely, we prove that determining the distant irregularity strength, where only vertices within a certain distance are required to be incident to different sums of labels, and the multiset irregularity strength, where any two distinct vertices are required to be incident to different multisets of labels, are NP-hard problems.