  The concept of low-congestion shortcuts is initiated by Ghaffari and Haeupler
[SODA2016] for addressing the design of CONGEST algorithms running fast in
restricted network topologies. Specifically, given a specific graph class $X$,
an $f$-round algorithm of constructing shortcuts of quality $q$ for any
instance in $X$ results in $\tilde{O}(q + f)$-round algorithms of solving
several fundamental graph problems such as minimum spanning tree and minimum
cut, for $X$.
  In this paper, we consider the relationship between the quality of
low-congestion shortcuts and three major graph parameters, chordality,
diameter, and clique-width. The main contribution of the paper is threefold:
(1) We show an $O(1)$-round algorithm which constructs a low-congestion
shortcut with quality $O(kD)$ for any $k$-chordal graph, and prove that the
quality and running time of this construction is nearly optimal up to
polylogarithmic factors. (2) We present two algorithms, each of which
constructs a low-congestion shortcut with quality $\tilde{O}(n^{1/4})$ in
$\tilde{O}(n^{1/4})$ rounds for graphs of $D=3$, and that with quality
$\tilde{O}(n^{1/3})$ in $\tilde{O}(n^{1/3})$ rounds for graphs of $D=4$
respectively. These results obviously deduce two MST algorithms running in
$\tilde{O}(n^{1/4})$ and $\tilde{O}(n^{1/3})$ rounds for $D=3$ and $4$
respectively, which almost close the long-standing complexity gap of the MST
construction in small-diameter graphs originally posed by Lotker et al.
[Distributed Computing 2006]. (3) We show that bounding clique-width does not
help the construction of good shortcuts by presenting a network topology of
clique-width six where the construction of MST is as expensive as the general
case.
