Connected tree-width

TL;DR

This paper characterizes graphs with small connected tree-width, linking it to small tree-width and absence of long geodesic cycles, and explores their hyperbolicity properties.

Contribution

It provides a new characterization of connected tree-width, establishes a duality theorem, and connects connected tree-width to hyperbolicity, extending understanding of graph structure.

Findings

Graphs of connected tree-width k are k-hyperbolic.

Graphs with bounded tree-width and short geodesic cycles are hyperbolic with a specific bound.

Connected tree-width is characterized by the absence of large brambles with connected covers.

Abstract

The connected tree-width of a graph is the minimum width of a tree-decomposition whose parts induce connected subgraphs. Long cycles are examples of graphs that have small tree-width but large connected tree-width. We show that a graph has small connected tree-width if and only if it has small tree-width and contains no long geodesic cycle. We further prove a connected analogue of the duality theorem for tree-width: a finite graph has small connected tree-width if and only if it has no bramble whose connected covers are all large. Both these results are qualitative: the bounds are good but not tight. We show that graphs of connected tree-width $k$ are $k$-hyperbolic, which is tight, and that graphs of tree-width $k$ whose geodesic cycles all have length at most $\ell$ are $\lfloor{3\over2}\ell(k-1)\rfloor$-hyperbolic. The existence of such a function $h(k,\ell)$ had been conjectured by Sullivan.