Abstract
We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given any graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices is in FPT parameterized by k on well-partitioned chordal graphs, while on chordal graphs, these problems are only known to be in XP. From the other end, we observe that there are problems that are polynomial-time solvable on split graphs, but become NP-complete on well-partitioned chordal graphs.
J. Ahn and O. Kwon are supported by the Institute for Basic Science (IBS-R029-C1). O. Kwon is also supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No. NRF-2018R1D1A1B07050294). L. Jaffke is supported by the Trond Mohn Foundation (TMS). P. T. Lima is supported by the Research Council of Norway via the project “CLASSIS”.
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Notes
- 1.
Note that holes in the sense of [18] are chordless cycles on at least five vertices; we can check for \(C_4\) separately by brute force. While there are algorithms that verify chordality more directly, we use this procedure to fulfil the promise that we can always output an obstruction if there is one.
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Ahn, J., Jaffke, L., Kwon, Oj., Lima, P.T. (2020). Well-Partitioned Chordal Graphs: Obstruction Set and Disjoint Paths. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_12
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