Maximum locally irregular induced subgraphs via minimum irregulators

Foivos Fioravantes, Nikolaos Melissinos, Theofilos Triommatis

Abstract: If a graph G is such that no two of its adjacent vertices have the same degree, we say that G is locally irregular. In this work we introduce and study the problem of finding a largest locally irregular induced subgraph of a given graph G. Equivalently, given a graph G, find a subset S of V(G) with minimum order, such that 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 examine some easy graph families, namely paths, cycles, 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 planar bipartite, or cubic bipartite graphs. We then show that we cannot even approximate an optimal solution within a ratio of O(n1−1k), where k≥1 and n is the order of the graph, unless P=NP, even when the input graph is bipartite.Then, looking for more positive results, we turn our attention towards computing 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 Δ of G with running time (2Δ)knO(1), while the second one considers the treewidth tw and Δ of G, and has running time Δ3twnO(1). Therefore, we show that the problem is in FPT by both k and tw if the graph has bounded maximum degree Δ.Since the algorithms we present are not in FPT for graphs with unbounded maximum degree (unless we consider Δ+k or Δ+tw as the parameter), it is natural to wonder if there exists an algorithm that does not include additional parameters (other than k or tw) in its dependency. We answer negatively to this question. In particular, we prove that there is no algorithm that computes I(G) with dependence f(k)no(k) or f(tw)no(tw), unless the ETH fails, showing that our algorithms are essentially optimal.