Directed Steiner path packing and directed path connectivity

TL;DR

This paper investigates the complexity and bounds of directed path connectivity measures in various classes of directed graphs, extending classical undirected graph connectivity concepts.

Contribution

It provides complexity results for path connectivity parameters on Eulerian, symmetric, and general digraphs, and establishes bounds for these parameters.

Findings

Complexity results for $ abla^p_{S,r}(D)$ on Eulerian and symmetric digraphs.

Complexity results for $ abla^p_{S,r}(D)$ on general digraphs.

Bounds for $ abla^p_k(D)$ and $ abla^p_k(D)$ parameters.

Abstract

For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, a directed $(S, r)$-Steiner path or, simply, an $(S, r)$-path is a directed path $P$ started at $r$ with $S\subseteq V(P)$. Two $(S, r)$-paths are said to be arc-disjoint if they have no common arc. Two arc-disjoint $(S, r)$-paths are said to be internally disjoint if the set of common vertices of them is exactly $S$. Let $\kappa^p_{S,r}(D)$ (resp. $\lambda^p_{S,r}(D)$) be the maximum number of internally disjoint (resp. arc-disjoint) $(S, r)$-paths in $D$. The directed path $k$-connectivity of $D$ is defined as $$\kappa^p_k(D)= \min \{\kappa^p_{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}.$$ Similarly, the directed path $k$-arc-connectivity of $D$ is defined as $$\lambda^p_k(D)= \min \{\lambda^p_{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}.$$ The directed path $k$-connectivity and directed path $k$-arc-connectivity are also called directed path connectivity which extends the path connectivity on undirected graphs to directed graphs and could be seen as a generalization of classical connectivity of digraphs. In this paper, we obtain complexity results for $\kappa^p_{S,r}(D)$ on Eulerian digraphs and symmetric digraphs, and $\lambda^p_{S,r}(D)$ on general digraphs. We also give bounds for the parameters $\kappa^p_k(D)$ and $\lambda^p_k(D)$.