Recognizing Proper Tree-Graphs
Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, Du\v{s}an Knop

TL;DR
This paper studies the complexity of recognizing proper H-graphs, showing fixed-parameter tractability when H is a tree and NP-completeness for certain multigraphs, thus clarifying the computational boundaries.
Contribution
It establishes fixed-parameter tractability for recognizing proper H-graphs when H is a tree and NP-completeness for some multigraphs, advancing understanding of their computational complexity.
Findings
Recognition is fixed-parameter tractable for proper T-graphs when T is a tree.
Recognition becomes NP-complete for certain multigraphs with 4 vertices and 5 edges.
Provides algorithms and complexity results for proper H-graph recognition.
Abstract
We investigate the parameterized complexity of the recognition problem for the proper -graphs. The -graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph , and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of -graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Bir\'o, Hujter, and Tuza in 1992, and the proper -graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. - For a tree with …
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