On Tree-Connectivity and Path-Connectivity of Graphs

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 $\kappa _k(G)\ge \ell$.