Graph parameters that are coarsely equivalent to tree-length

Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length of a tree-decomposition ${\cal T}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal T}(G)$, and the tree-length of $G$ is the minimum of the length, over all tree-decompositions of $G$. We present simpler and sometimes with better bounds proofs for those known in literature results and further extend this list of graph parameters coarsely equivalent to tree-length. Among other new results, we show that the tree-length of a graph $G$ is small if and only if for every bramble ${\cal F}$ (or every Helly family of connected subgraphs ${\cal F}$, or every Helly family of paths ${\cal F}$) of $G$, there is a disk in $G$ with small radius that intercepts all members of ${\cal F}$. Furthermore, the tree-length of a graph $G$ is small if and only if $G$ can be embedded with a small additive distortion to an unweighted tree with the same vertex set as in $G$ (not involving any Steiner points). Additionally, we introduce a new natural "bridging`` property for cycles, which generalizes a known property of cycles in chordal graphs, and show that it also coarsely defines the tree-length.