On Tree-Connectivity and Path-Connectivity of Graphs

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 .