Parameterized Aspects of Strong Subgraph Closure
| Autor: | Athanasios L. Konstantinidis, Petr A. Golovach, Pinar Heggernes, Paloma T. Lima, Charis Papadopoulos |
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| Rok vydání: | 2018 |
| Předmět: |
FOS: Computer and information sciences
General Computer Science Discrete Mathematics (cs.DM) Generalization Induced subgraph Parameterized complexity 0102 computer and information sciences 02 engineering and technology 01 natural sciences Combinatorics Polynomial kernel Computer Science - Data Structures and Algorithms 0202 electrical engineering electronic engineering information engineering Data Structures and Algorithms (cs.DS) Split graph Mathematics 000 Computer science knowledge general works Applied Mathematics Computer Science Applications Triadic closure 010201 computation theory & mathematics Theory of computation Computer Science 020201 artificial intelligence & image processing Relaxation (approximation) Computer Science - Discrete Mathematics |
| DOI: | 10.48550/arxiv.1802.10386 |
| Popis: | Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the StrongF-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In StrongF-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence, the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to forbid that strong copy of F in G. We study StrongF-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when $$F =P_3$$. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization $$k - \mu (G)$$, where $$\mu (G)$$ is the maximum matching size of G. We conclude with some results on the parameterization of StrongF-closure by the number of edges of G that are not selected as strong. |
| Databáze: | OpenAIRE |
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