Well-partitioned chordal graphsopen access
- Authors
- Ahn, Jungho; Jaffke, Lars; Kwon, O-joung; Lima, Paloma T.
- Issue Date
- Oct-2022
- Publisher
- ELSEVIER
- Keywords
- Well-partitioned chordal graph; Forbidden induced subgraphs; Graph class; Longest path transversal; Tree spanner; Geodetic set
- Citation
- DISCRETE MATHEMATICS, v.345, no.10, pp.1 - 23
- Indexed
- SCIE
SCOPUS
- Journal Title
- DISCRETE MATHEMATICS
- Volume
- 345
- Number
- 10
- Start Page
- 1
- End Page
- 23
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/171438
- DOI
- 10.1016/j.disc.2022.112985
- ISSN
- 0012-365X
- Abstract
- We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X, Y is an element of P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C, I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We observe that there are problems, for instance DENSEST k-SUBGRAPH and b-COLORING, that are polynomial-time solvable on split graphs but become NP-hard on well-partitioned chordal graphs. On the other hand, we show that the GEODETIC SET problem, known to be NP-hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs. We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles). (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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