Remoteness, order, size and connectivity constraints in digraphs
Sufiyan Mallu
TL;DR
This work addresses the problem of determining the maximum remoteness in strong digraphs under constraints on order, size, and connectivity. It introduces κ-connected path-complete digraphs and proves that the extremal structure achieving the maximal remoteness is the digraph $DPK_{n,m,κ}$, yielding a sharp bound $\rho(D) \le \rho(DPK_{n,m,κ})$ with equality characterized by a modular condition on $m$; the bound is given explicitly as $\rho(DPK_{n,m,κ}) = \frac{n}{κ} + 2 - \frac{1}{κ} - \frac{κ-1}{n-1} - \frac{m^*}{κ(n-1)}$. A special case $κ=1$ reduces to $\rho(D) \le n+1 - \frac{m}{n-1}$. The paper then extends these graph bounds to Eulerian digraphs, showing that the same extremal framework yields sharp bounds for Eulerian digraphs with given order and size, and also for $\lambda$-edge-connected cases with $\lambda \in \{2,3\}$, by relating to corresponding (undirected) graph bounds. Overall, the results bridge undirected remoteness theory and directed distance invariants, delivering exact extremal structures and broad applicability to Eulerian digraphs and connectivity-constrained families.
Abstract
Let \( D \) be a strongly connected digraph. The average distance of a vertex \( v \) in \( D \) is defined as the arithmetic mean of the distances from \( v \) to all other vertices in \( D \). The remoteness \( ρ(D) \) of \( D \) is the maximum of the average distances of the vertices in \( D \). In this paper, we provide a sharp upper bound on the remoteness of a strong digraph with given order, size, and vertex-connectivity. We then characterise the extremal digraphs that maximise remoteness among all strong digraphs of order \(n\), size at least \(m\), and vertex-connectivity \(κ\). Finally, we demonstrate that the upper bounds on the remoteness of a graph given its order, size, and connectivity constraints (see \cite{DanMafMal2025}) can be extended to a larger class of digraphs containing all graphs, the Eulerian digraphs.