On Tree-Connectivity and Path-Connectivity of Graphs (original) (raw)

Abstract

Let G be a graph and k an integer with \(2\le k\le n\). The k -tree-connectivity of G, denoted by \(\kappa _k(G)\), is defined as the minimum \(\kappa _G(S)\) over all _k_-subsets S of vertices, where \(\kappa _G(S)\) denotes the maximum number of internally disjoint _S_-trees in G. The k -path-connectivity of G, denoted by \(\pi _k(G)\), is defined as the minimum \(\pi _G(S)\) over all _k_-subsets S of vertices, where \(\pi _G(S)\) denotes the maximum number of internally disjoint _S_-paths in G. In this paper, we first prove that for any fixed integer \(k\ge 1\), given a graph G and a subset S of V(G), deciding whether \(\pi _G(S)\ge k\) is \(\mathcal {NP}\)-complete. Moreover, we also show that for any fixed integer \(k_1\ge 5\), given a graph G, a \(k_1\)-subset S of V(G) and an integer \(1\le k_2\le n-1\), deciding whether \(\pi _G(S)\ge k_2\) is \(\mathcal {NP}\)-complete. Let \(\pi (k,\ell )=1+\max \{\kappa (G)|\ \)G\(\text {\ is\ a\ graph\ with}\ \pi _k(G)< \ell \}\). Hager (Discrete Math 59:53–59, 1986) showed that \(\ell (k-1)\le \pi (k,\ell )\le 2^{k-2}\ell\) and conjectured that \(\pi (k,\ell )=\ell (k-1)\) for \(k\ge 2\) and \(\ell \ge 1\). He also confirmed the conjecture for \(2\le k\le 4\) and proved \(\pi (5,\ell )\le \lceil \frac{9}{2}\ell \rceil\). By introducing a “Generalized Path-Bundle Transformation”, we confirm the conjecture for \(k=5\) and prove that \(\pi (k,\ell )\le 2^{k-3}\ell\) for \(k\ge 5\) and \(\ell \ge 1\). By employing this transformation, we also prove that if G is a graph with \(\kappa (G)\ge (k-1)\ell\) for any \(k\ge 2\) and \(\ell \ge 1\), then \(\kappa _k(G)\ge \ell\).

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Acknowledgements

Shasha Li was supported by Zhejiang Provincial Natural Science Foundation of China (No. LY18A010002), and the Natural Science Foundation of Ningbo, China (No. 202003N4148). Zhongmei Qin was partially supported by National Natural Science Foundation of China (No. 11901050) and Natural Science Basic Research Program of Shaanxi (Nos. 2020JQ–336, 2021JQ–219). Jianhua Tu was partially supported by Research Foundation for Advanced Talents of Beijing Technology and Business University. Jun Yue was partially supported by the National Natural Science Foundation of China (No. 11701342) and the Natural Science Foundation of Shandong Province (No. ZR2016AQ01).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, Zhejiang, People’s Republic of China
    Shasha Li
  2. School of Science, Chang’an University, Xi’an, 710064, Shaanxi, People’s Republic of China
    Zhongmei Qin
  3. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, People’s Republic of China
    Jianhua Tu
  4. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, Shandong, People’s Republic of China
    Jun Yue

Authors

  1. Shasha Li
  2. Zhongmei Qin
  3. Jianhua Tu
  4. Jun Yue

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Correspondence toZhongmei Qin.

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Li, S., Qin, Z., Tu, J. et al. On Tree-Connectivity and Path-Connectivity of Graphs.Graphs and Combinatorics 37, 2521–2533 (2021). https://doi.org/10.1007/s00373-021-02376-9

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