Strong triadic closure in cographs and graphs of low maximum degree

Abstract

The MaxSTC problem is an assignment of the edges with two types of labels, namely, strong and weak, that maximizes the number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent. The Cluster Deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques. Both problems are known to be NP-hard and an optimal solution for the Cluster Deletion problem provides a feasible solution for the MaxSTC problem, however not necessarily an optimal one. In this work we conduct the first systematic study that reveals graph families for which the optimal solutions for MaxSTC and Cluster Deletion coincide. We first show that MaxSTC coincides with Cluster Deletion on cographs and, thus, MaxSTC is solvable in polynomial time on cographs. As a side result, we give an interesting computational characterization of the maximum independent set on the cartesian product of two cographs. Furthermore, we address the influence of the low degree bounds to the complexity of the MaxSTC problem. We show that this problem is polynomial-time solvable on graphs of maximum degree three, whereas MaxSTC becomes NP-complete on graphs of maximum degree four. The proof of the latter result implies that there is no subexponential-time algorithm for MaxSTC unless the Exponential-Time Hypothesis fails.