Treewidth
Treewidth^{15} is a metric that can be used to measure the complexity of a graph and is based on the notion that many hard problems are actually extremely easy to solve on trees. Treewidth essentially says something about how close a graph is to being a tree. The notion of treewidth is extremely relevant to CPQs as CPQs have a treewidth of 2, this means that fixed parameter tractable (FTP) algorithms that have a runtime exponential in the treewidth can be viable solutions to problems involving CPQs.
However, one problem with treewidth is that it goes by many different names, the following list attempts to give an exhaustive overview of definitions used to refer to treewidth (or definitions later discovered to be equivalent to treewidth). It is worth noting that in order to make the treewidth of a tree 1, some definitions subtract 1 from the computed complexity value (e.g. a minimum width tree decomposition of a tree has bags of size 2, but the treewidth is 1).

The minimum width of a tree decomposition^{6}. This is the size of the largest bag in the a tree decomposition with minimum bag size minus one.

The minimum clique number of a chordal supergraph^{6}. Or alternatively^{4}, if $G$ has treewidth at most $k$ then $G$ is a subgraph of a chordal graph with maximum clique size $k + 1$.

The minimum cost of an elimination order^{6}.

The minimum $k$ for which a graph is a partial $k$tree^{6}. Or alternatively^{4}, if $G$ has treewidth at most $k$ then $G$ is a partial $k$tree.

By extension, a partial $k$tree has treewidth $k$.

The minimum number of cops needed to catch a visible robber on a graph (minus one)^{6}. This is based on the notion of havens where if a graph has a maximum order haven of order $k + 1$ then it has treewidth $k$.

In a family of connected subgraphs that all touch each other (vertex or edge shared). The maximum order of a bramble minus one, where the order of a bramble is the smallest hitting set that hits all the family subgraphs^{7}.

The dimension of a graph by Bertelè and Brioschi^{1,5} (not to be confused with the dimension of a graph as defined by Erdős, Harary and Tutte^{2}). If the treewidth of $G$ is at most $k$ then the treewdith of $G$ is at most $k$^{4}.

As defined in Halin’s study of Sfunctions^{3,5}.

If $G$ has treewidth at most $k$ then $G$ is $k$decomposable^{4}.

If $G$ has treewidth at most $k$ then $G$ has no screen of thickness at least $k + 2$^{4}.

If $G$ has treewidth at most $k$ then $k + 1$ cops can search $G$ in the SeymourThomas search game^{4}.

If $G$ has treewidth at most $k$ then $k + 1$ cops can monotonely search $G$ in the SeymourThomas search game^{4}.

If $G$ has treewidth at most $k$ then the number of searchers, needed to search $G$ in the fugitive search game with an inert fugitive is at most $k + 1$^{4}.

If $G$ has treewidth at most $k$ then the number of searchers, needed to monotonically search $G$ in the fugitive search game with an inert fugitive is at most $k + 1$^{4}.
For a lot of useful properties of graphs of bounded treewidth (and similar notions) the survey paper by H. L. Bodlaender is a good start^{4}.
References:
 U. Bertelè and F. Brioschi. Nonserial Dynamic Programming. Academic Press, New York, 1972.
 P. Erdős, F. Harary, and W. T. Tutte. On the dimension of a graph. Mathematika 12:118–122, 1965.
 R. Halin. Sfunctions for graphs. J. Geom. 8(1–2):171–186, 1976.
 H. L. Bodlaender. A partial karboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209:1–45, 1998.
 Lecture 11 “Treewidth” from 2WO08 Computational Topology by Jeff Erickson (2015)
 Lecture 9 “Treewidth” from 2IMA25 Exact Algorithms for NPhard Problems by Bart Jansen (2022)
 Wikipedia on Treewidth
 Wikipedia on Tree decomposition
 Wikipedia on Chordal completion
 Wikipedia on Partial ktree
 Wikipedia on Haven
 Wikipedia on Bramble
 Wikipedia on Set cover problem
 MIT Advanced Algorithms Lecture 15 on treewidth and elimination orderings by David Karger (2004)
 Neil Robertson and P.D Seymour. Graph minors. ii. algorithmic aspects of treewidth. Journal of Algorithms, 1986.