Maximizing the Strong Triadic Closure in Split Graphs and Proper Interval Graphs

TL;DR

This paper investigates the computational complexity of maximizing strong edges under the Strong Triadic Closure constraint in different graph classes, providing polynomial algorithms for some and NP-completeness results for others.

Contribution

It identifies specific graph classes where the problem is polynomial-time solvable and establishes NP-completeness for split graphs, delineating the boundary between tractable and intractable cases.

Findings

Polynomial-time algorithm for proper interval graphs

Polynomial-time algorithm for trivially-perfect graphs

NP-completeness for split graphs

Abstract

In social networks the {\sc Strong Triadic Closure} is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of strong edges that satisfy the strong triadic closure was recently shown to be NP-complete for general graphs. Here we initiate the study of graph classes for which the problem is solvable. We show that the problem admits a polynomial-time algorithm for two unrelated classes of graphs: proper interval graphs and trivially-perfect graphs. To complement our result, we show that the problem remains NP-complete on split graphs, and consequently also on chordal graphs. Thus we contribute to define the first border between graph classes on which the problem is polynomially solvable and on which it remains NP-complete.