Abstract
In social networks the 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 incomparable 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.
Highlights
Predicting the behavior of a network is an important concept in the field of social networks [9]
Understanding the strength and nature of social relationships has found an increasing usefulness in the last years due to the explosive growth of social networks. Towards such a direction the Strong Triadic Closure principle enables us to understand the structural properties of the underlying graph: it is not possible for two individuals to have a strong relationship with a common friend and not know each other [12]
Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 53:2 Strong Triadic Closure in Split Graphs and Proper Interval Graphs no induced path on three vertices which is equivalent with a graph that consists of vertexdisjoint union of cliques) there is a trivial solution by labeling strong all the edges
Summary
Predicting the behavior of a network is an important concept in the field of social networks [9]. 53:2 Strong Triadic Closure in Split Graphs and Proper Interval Graphs no induced path on three vertices which is equivalent with a graph that consists of vertexdisjoint union of cliques) there is a trivial solution by labeling strong all the edges. Such an observation might falsely lead into a graph modification problem, known as Cluster Deletion problem (see e.g., [3, 14]), in which we want to remove the minimum number of edges that correspond to the weak edges, such that the resulting graph does not contain a P3 as an induced subgraph. By considering the characterization of the induced P3’s mentioned earlier, we show that MaxSTC admits a simple polynomial-time solution on trivially-perfect graphs (i.e., graphs having no induced P4 or C4)
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