Bridge distances for networks

Let $G = (V,E)$ be a finite directed graph with a non-negative real length $\mu_e$ assigned to every directed edge $e \in E$. We assume that $\mu_e = +\infty$ for every non-edge $e \not\in E$. Fix any two distinct vertices $a, b \in V$. A directed path from $a$ to $b$ is called an $(a,b)$-path. An edge $e$ is called an $(a,b)$-bridge if it belongs to all $(a,b)$-paths. Furthermore, it is not difficult to show that all $(a,b)$-paths pass all $(a,b)$-bridges in the same order. Define the distance $\mu(a,b)$ from $a$ to $b$ as the sum of lengths of all $(a,b)$-bridges. Furthermore, $\mu(a,b) = \infty$ if there are no $(a,b)$-paths and $\mu(a,b) = 0$ if $(a,b)$-paths exist but there are no $(a,b)$-bridges. It is easily seen that $\mu(a,b)$ can be computed in polynomial time and the metric inequality $\mu(a,b) \leq \mu(a,c) + \mu(c,b)$ holds for every $a,b,c \in V$. Furthermore, equality holds if and only if each $(a,b)$-bridge is either an $(a,c)$- or a $(c,b)$-bridge. \newline We will show that this is a special limit case $r=s \rightarrow 0$ of the inequality $\mu(a,b)^{s/r} \leq \mu(a,c)^{s/r} + \mu(c,b)^{s/r}$ obtained for all positive real parameters $r$ and $s$ in the paper ``Metric and ultrametric inequalities for directed graphs'', Discrete Appl. Math. 314 (2022) 93--104, along with 3 other limit cases $r=s \rightarrow \infty$, $r=1, s \rightarrow \infty$, and $s = 1, r \rightarrow 0$, considered in that paper.