Correcting a Graph Into a Linegraph Minimizing Hamming Distance Edition Is NP-Complete and FPT by Treewidth
Year: 2025
Type: article
Abstract: Since Beineke's work in 1968 on linegraphs, attention has focused on the classification of graphs as linegraphs. It is known that every graph $G$ is the linegraph of an hypergraph, and the question is to characterize that root graph. We introduce the $C_{p,q}$ classes, defined as sets of graphs where each vertex can be covered by at most $p$ cliques, and each edge belongs to at most $q$ cliques. These classes provide a comprehensive classification of linegraphs through a unified and parameterized approach. They describe previously known graph classes - such as linegraphs of simple graphs, $p$-uniform hypergraphs and $p$-uniform $1$-linear hypergraphs - while being capable of generalization. We study the complexity of determining the membership and edit distance of a graph to one of these classes. We prove the first Fixed Parameter Tractable algorithm with respect to treewidth to compute the edit distance.
Institutions: Université de Versailles Saint-Quentin-en-Yvelines, Données et algorithmes pour une ville intelligente et durable, Services répartis, Architectures, MOdélisation, Validation, Administration des Réseaux, Université Paris-Saclay, CentraleSupélec, Laboratoire Interdisciplinaire des Sciences du Numérique
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409753074