1. Precedence thinness in graphs
- Author
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Jayme Luiz Szwarcfiter, Moysés S. Sampaio, Fabiano de S. Oliveira, and Flavia Bonomo-Braberman
- Subjects
FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,05C75, 05C85 ,Applied Mathematics ,G.2.2 ,Characterization (mathematics) ,Graph ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Interval (graph theory) ,Combinatorics (math.CO) ,Recognition algorithm ,Time complexity ,Computer Science - Discrete Mathematics ,Mathematics - Abstract
Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of $k$ interval graphs, by splitting the graph in some special way. As a recent example of such an approach, the classes of $k$-thin and proper $k$-thin graphs have been introduced generalizing interval and proper interval graphs, respectively. The complexity of the recognition of each of these classes is still open, even for fixed $k \geq 2$. In this work, we introduce a subclass of $k$-thin graphs (resp. proper $k$-thin graphs), called precedence $k$-thin graphs (resp. precedence proper $k$-thin graphs). Concerning partitioned precedence $k$-thin graphs, we present a polynomial time recognition algorithm based on $PQ$-trees. With respect to partitioned precedence proper $k$-thin graphs, we prove that the related recognition problem is \NP-complete for an arbitrary $k$ and polynomial-time solvable when $k$ is fixed. Moreover, we present a characterization for these classes based on threshold graphs., 33 pages
- Published
- 2022