Abstract (EN)

If a graph G is such that no two adjacent vertices of G have the same degree, we say that G is locally irregular. In this work we introduce and study the problem of identifying a largest induced subgraph of a given graph G that is locally irregular. Equivalently, given a graph G, find a subset S of V(G) of minimum order, such that by deleting the vertices of S from G results in a locally irregular graph; we denote with I(G) the order of such a set S. We first treat some easy graph families, namely paths, cycles, trees, complete bipartite and complete graphs. However, we show that the decision version of the introduced problem is NP-Complete, even for restricted families of graphs, such as subcubic bipartite, or cubic graphs. Then, looking for more positive results, we turn towards computing the parameter I(G) through the lens of parameterised complexity. In particular, we provide two algorithms that compute I(G), each one considering different parameters. The first one considers the size of the solution k and the maximum degree ∆ of G with running time (2∆)^k n^ O(1) , while the second one considers the treewidth tw and ∆ of G, and has running time ∆ ^(2tw)n^O(1). Therefore, we show that the problem is FPT by both k and tw if the graph has bounded maximum degree ∆. Since these algorithms are not FPT for graphs with unbounded maximum degree (unless we consider ∆ + k or ∆ + tw as the parameter), it is natural to wonder about the existence of an algorithm that does not include additional parameters (other than k or tw) in its dependency. We manage to settle negatively this question, and we show that our algorithms are essentially optimal. In particular, we prove that there is no algorithm that computes I(G) with dependence f (k)n^o(k) or f (tw)n^o(tw) , unless the ETH fails.