Your rugby mates don't need to know your colleagues: Triadic closure with edge colors

Abstract Given an undirected graph G = ( V , E ) the NP-hard Strong Triadic Closure (STC) problem asks for a labeling of the edges as weak and strong such that at most k edges are weak and for each induced P 3 in G at least one edge is weak. We study the following generalizations of STC with c different strong edge colors. In Multi-STC an induced P 3 may receive two strong labels as long as they are different. In Edge-List Multi-STC and Vertex-List Multi-STC we may restrict the set of permitted colors for each edge of G. We show that, under the Exponential Time Hypothesis (ETH), Edge-List Multi-STC and Vertex-List Multi-STC cannot be solved in time 2 o ( | V | 2 ) . We proceed with a parameterized complexity analysis in which we extend previous algorithms and kernelizations for STC [Golovach et al., Algorithmica'20, Gruttemeier and Komusiewicz, Algorithmica'20] to the three variants or outline the limits of such an extension.