Complexity of Finding Maximum Locally Irregular Induced Subgraphs 1

Maximising 1’s through proper labellings

“…These ideas can be implemented through the concept of irregulators, introduced in [8]. Recall that a locally irregular graph is a graph in which adjacent vertices have different degrees.…”

“…Now, for a graph G, a set S ⊂ V (G) of vertices is called an irregulator if G − S is locally irregular. In [8], the authors investigated the problem of determining the smallest irregulator of a given graph, which parameter we denote by Irr(G) for a given graph G. Note that determining Irr(G) Next we consider the vertices of T in reverse order of their distance to r in T . Whenever considering a new vertex v this way, this means there remains only one incident edge vu in T to be labelled (where u is closer to r than v is), together, possibly, with edges incident to v going to H. We first assign labels to the edges incident to v going to H so that no conflicts involving the neighbours of v arise (which can be done as previously).…”

“…On a different note, while finding, in a given graph, a largest induced subgraph satisfying some property is a classical problem, [8] did not report concrete applications or problems where finding small irregulators would be useful. To the best of our knowledge, our use of these notions is thus one of its very first applications.…”

“…J. Bensmail is equivalent to finding the largest locally irregular induced subgraph of G. The authors of [8] mainly investigated the general complexity of determining Irr(G) for a given graph G, establishing, on the negative side, NP-hardness results and non-approximation results on the problem. On the positive side, they proved that Irr(G) can be determined in polynomial time provided G is a path, a cycle, a tree, a complete bipartite graph, or a complete graph, and provided FPT results.…”