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Graph Searches and Their End Vertices

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Abstract

For a graph search algorithm, the end vertex problem is concerned with which vertices of a graph can be the last visited by this algorithm. We characterize all maximum cardinality searches on chordal graphs and derive from this characterization a polynomial-time algorithm for the end vertex problem of maximum cardinality searches on chordal graphs. It is complemented by a proof of NP-completeness of the same problem on weakly chordal graphs. We also show linear-time algorithms for deciding end vertices of breadth-first searches on interval graphs and end vertices of lexicographic depth-first searches on chordal graphs.

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Notes

  1. A graph is a split graph if its vertex set can be partitioned into a clique and an independent set. Definitions of weakly chordal graphs and interval graphs are deferred to Sects. 4 and 6, respectively.

  2. The algorithm mcs\(^{+}\) is defined in the same spirit of lbfs\(^{+}\), and it has not been explicitly mentioned in literature because no application of this algorithm has been discovered.

  3. This is not a problem for lbfs\(^{+}\), because it never merges two parts.

  4. One may note that \(\{[{\mathtt {lp}(v)}, {\mathtt {rp}(v)}]\mid v\in V(G)\}\) gives an interval representation for G.

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Acknowledgements

Y.C. would like to thank Jing Huang for bringing the end vertex problems to his attention.

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Authors and Affiliations

  1. School of Computer Science and Engineering, Central South University, Changsha, China

    Guozhen Rong, Yixin Cao, Jianxin Wang & Zhifeng Wang

  2. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

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Correspondence to Yixin Cao.

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A preliminary version of this paper appears in the Proceedings of the 30th International Symposium on Algorithms and Computation (ISAAC 2019).

G. Rong, Y. Cao, J. Wang, Z. Wang: Supported by National Natural Science Foundation of China (61828205, 61672536), Hunan Provincial Key Lab on Bioinformatics, and Hunan Provincial Science and Technology Program (2018WK4001).

Y. Cao: Supported in part by the Hong Kong Research Grants Council (RGC) under Grant 15201317 and the National Natural Science Foundation of China (NSFC) under Grant 61972330.

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Rong, G., Cao, Y., Wang, J. et al. Graph Searches and Their End Vertices. Algorithmica 84, 2642–2666 (2022). https://doi.org/10.1007/s00453-022-00981-5

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