Abstract
When can we compute the diameter of a graph in quasi linear time? We address this question for the class of split graphs, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either 2 or 3, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of a split graph in less than quadratic time. Therefore it is worth to study the complexity of diameter computation on subclasses of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded clique-interval number and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants and then almost completely settle the complexity of diameter computation on these subclasses of split graphs:
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For the k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if \(k=\mathcal{O}(1)\), and even in quasi linear time if \(k=o(\log {n})\) and in addition a corresponding ordering is given. However, under SETH this cannot be done in truly subquadratic time for any \(k = \omega (\log {n})\).
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For the complements of k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if \(k=\mathcal{O}(1)\), and even in time \(\mathcal{O}(km)\) if a corresponding ordering is given. Again this latter result is optimal under SETH up to polylogarithmic factors.
Our findings raise the question whether a k-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for \(k=1\) and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs – including the ones mentioned above – have a bounded clique-interval number. A research report version is deposited on HAL repository with number hal-02307397.
G. Ducoffe—This work was supported by a grant of Romanian Ministry of Research and Innovation CCCDI-UEFISCDI. Project no. 17PCCDI/2018.
M. Habib—Supported by Inria Gang project-team, and ANR project DISTANCIA (ANR-17-CE40-0015).
L. Viennot—Supported by Irif laboratory from CNRS and Paris University, and ANR project Multimod (ANR-17-CE22-0016).
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Notes
- 1.
This sparse representation is also called the neighbourhood set system of the stable set of G, see Sect. 2.
- 2.
We observe that the general graphs with bounded interval number are sometimes called “split interval” [2], that may create some confusion.
- 3.
We refer to [13] for a more general presentation of range trees.
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Ducoffe, G., Habib, M., Viennot, L. (2019). Fast Diameter Computation Within Split Graphs. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_13
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