Let G be a graph and k an integer with $$2\le k\le n$$ 2 ≤ k ≤ n . The k -tree-connectivity of G , denoted by $$\kappa _k(G)$$ κ k ( G ) , is defined as the minimum $$\kappa _G(S)$$ κ G ( S ) over all k -subsets S of vertices, where $$\kappa _G(S)$$ κ G ( S ) denotes the maximum number of internally disjoint S -trees in G . The k -path-connectivity of G , denoted by $$\pi _k(G)$$ π k ( G ) , is defined as the minimum $$\pi _G(S)$$ π G ( S ) over all k -subsets S of vertices, where $$\pi _G(S)$$ π G ( S ) denotes the maximum number of internally disjoint S -paths in G . In this paper, we first prove that for any fixed integer $$k\ge 1$$ k ≥ 1 , given a graph G and a subset S of V ( G ), deciding whether $$\pi _G(S)\ge k$$ π G ( S ) ≥ k is $$\mathcal {NP}$$ NP -complete. Moreover, we also show that for any fixed integer $$k_1\ge 5$$ k 1 ≥ 5 , given a graph G , a $$k_1$$ k 1 -subset S of V ( G ) and an integer $$1\le k_2\le n-1$$ 1 ≤ k 2 ≤ n - 1 , deciding whether $$\pi _G(S)\ge k_2$$ π G ( S ) ≥ k 2 is $$\mathcal {NP}$$ NP -complete. Let $$\pi (k,\ell )=1+\max \{\kappa (G)|\ $$ π ( k , ℓ ) = 1 + max { κ ( G ) | G $$\text {\ is\ a\ graph\ with}\ \pi _k(G)< \ell \}$$ \ is\ a\ graph\ with π k ( G ) < ℓ } . Hager (Discrete Math 59:53–59, 1986) showed that $$\ell (k-1)\le \pi (k,\ell )\le 2^{k-2}\ell$$ ℓ ( k - 1 ) ≤ π ( k , ℓ ) ≤ 2 k - 2 ℓ and conjectured that $$\pi (k,\ell )=\ell (k-1)$$ π ( k , ℓ ) = ℓ ( k - 1 ) for $$k\ge 2$$ k ≥ 2 and $$\ell \ge 1$$ ℓ ≥ 1 . He also confirmed the conjecture for $$2\le k\le 4$$ 2 ≤ k ≤ 4 and proved $$\pi (5,\ell )\le \lceil \frac{9}{2}\ell \rceil$$ π ( 5 , ℓ ) ≤ ⌈ 9 2 ℓ ⌉ . By introducing a “Generalized Path-Bundle Transformation”, we confirm the conjecture for $$k=5$$ k = 5 and prove that $$\pi (k,\ell )\le 2^{k-3}\ell$$ π ( k , ℓ ) ≤ 2 k - 3 ℓ for $$k\ge 5$$ k ≥ 5 and $$\ell \ge 1$$ ℓ ≥ 1 . By employing this transformation, we also prove that if G is a graph with $$\kappa (G)\ge (k-1)\ell$$ κ ( G ) ≥ ( k - 1 ) ℓ for any $$k\ge 2$$ k ≥ 2 and $$\ell \ge 1$$ ℓ ≥ 1 , then $$\kappa _k(G)\ge \ell$$ κ k ( G ) ≥ ℓ .
[1]
G. Dirac.
In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen†
,
1960
.
[2]
Rong-Xia Hao,et al.
The generalized connectivity of alternating group graphs and (n, k)-star graphs
,
2018,
Discret. Appl. Math..
[3]
Xueliang Li,et al.
Note on the hardness of generalized connectivity
,
2012,
J. Comb. Optim..
[4]
J. A. Bondy,et al.
Graph Theory
,
2008,
Graduate Texts in Mathematics.
[5]
Yingbin Ma,et al.
Steiner tree packing number and tree connectivity
,
2018,
Discret. Math..
[6]
Xueliang Li,et al.
Sharp bounds for the generalized connectivity kappa3(G)
,
2009,
Discret. Math..
[7]
H. Whitney.
Congruent Graphs and the Connectivity of Graphs
,
1932
.
[8]
Michael Hager,et al.
Path-connectivity in graphs
,
1986,
Discret. Math..
[9]
Neil Robertson,et al.
Graph Minors .XIII. The Disjoint Paths Problem
,
1995,
J. Comb. Theory B.
[10]
Michael Hager,et al.
Pendant tree-connectivity
,
1985,
J. Comb. Theory, Ser. B.
[11]
E. L. Wilson,et al.
A family of path properties for graphs
,
1972
.
[12]
G. Chartrand,et al.
Rainbow trees in graphs and generalized connectivity
,
2010
.