  A classic 1993 paper by Alth\H{o}fer et al. proved a tight reduction from
spanners, emulators, and distance oracles to the extremal function $\gamma$ of
high-girth graphs. This paper initiated a large body of work in network design,
in which problems are attacked by reduction to $\gamma$ or the analogous
extremal function for other girth concepts. In this paper, we introduce and
study a new girth concept that we call the bridge girth of path systems, and we
show that it can be used to significantly expand and improve this web of
connections between girth problems and network design. We prove two kinds of
results:
  1) We write the maximum possible size of an $n$-node, $p$-path system with
bridge girth $>k$ as $\beta(n, p, k)$, and we write a certain variant for
"ordered" path systems as $\beta^*(n, p, k)$. We identify several arguments in
the literature that implicitly show upper or lower bounds on $\beta, \beta^*$,
and we provide some polynomially improvements to these bounds. In particular,
we construct a tight lower bound for $\beta(n, p, 2)$, and we polynomially
improve the upper bounds for $\beta(n, p, 4)$ and $\beta^*(n, p, \infty)$.
  2) We show that many state-of-the-art results in network design can be
recovered or improved via black-box reductions to $\beta$ or $\beta^*$.
Examples include bounds for distance/reachability preservers, exact hopsets,
shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an
integrality gap for directed Steiner forest.
  We believe that the concept of bridge girth can lead to a stronger and more
organized map of the research area. Towards this, we leave many open problems,
related to both bridge girth reductions and extremal bounds on the size of path
systems with high bridge girth.

Data Structures and Algorithms;Combinatorics