Diameter of orientations of graphs with given order and number of blocks

TL;DR

This paper establishes sharp upper bounds on the oriented diameter of bridgeless graphs based on their order and block structure, improving understanding of graph orientations with connectivity constraints.

Contribution

It provides new sharp bounds on the oriented diameter for graphs with given order and block count, including special cases like block graphs.

Findings

Bound of n - floor(p/2) for bridgeless graphs with p blocks

Sharpness of the bound for all n and p ≥ 2

Upper bound of approximately 3n/4 for bridgeless block graphs

Abstract

A strong orientation of a graph $G$ is an assignment of a direction to each edge such that $G$ is strongly connected. The oriented diameter of $G$ is the smallest diameter among all strong orientations of $G$. A block of $G$ is a maximal connected subgraph of $G$ that has no cut vertex. A block graph is a graph in which every block is a clique. We show that every bridgeless graph of order $n$ containing $p$ blocks has an oriented diameter of at most $n-\lfloor \frac{p}{2} \rfloor$. This bound is sharp for all $n$ and $p$ with $p \geq 2$. As a corollary, we obtain a sharp upper bound on the oriented diameter in terms of order and number of cut vertices. We also show that the oriented diameter of a bridgeless block graph of order $n$ is bounded above by $\lfloor \frac{3n}{4} \rfloor$ if $n$ is even and $\lfloor \frac{3(n+1)}{4} \rfloor$ if $n$ is odd.