Connected Tree-Width

Abstract

The connected tree-width of a graph is the minimum width of a treedecomposition 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 are 3 2 (k − 1) -hyperbolic. The existence of such a function h(k, ) had been conjectured by Sullivan. arXiv:1211.7353v3 [math.CO]

Mixing Time Bounds via Bottleneck Sequences

“…A graph is roughly isometric to a tree if and only if it has bounded connected treewidth. We refer to [4] for an explanation of tree-width. In considering whether Theorem 1.1 is best possible, it is natural to ask whether we might be able to say that every graph of bounded tree-width has mixing time within a constant factor of its tree decomposition, without requiring the parts of the tree decomposition to be connected.…”

Bounding Connected Tree-Width

“…We will now use ideas from [2] to bound the length of t-admissible paths in stable tree-decompositions. Together with Proposition 11, this will imply our main result.…”

“…The connected tree-width ctw(G) is defined accordingly as the minimum width of a connected tree-decomposition of the graph G. Trivially, the connected tree-width of a graph is at least as large as its tree-width and, as Jegou and Terrioux [5] observed, long cycles are examples of graphs of small tree-width but large connected tree-width. Diestel and Müller [2] showed that, more generally, the existence of long geodesic cycles, that is, cycles in a graph G that contain a shortest path in G between any two of their vertices, raises the connected treewidth. Furthmore, they proved that these two obstructions to small connected tree-width, namely, large tree-width and long geodesic cycles, are essentially the only obstructions: Theorem 1 ([2, Theorem 1.1]).…”

“…The tree-width duality theorem of Seymour and Thomas [6] asserts that a graph has tree-width less than k if and only if it has no bramble of order at least k. Diestel and Müller [2] conjectured that a similar duality holds for connected tree-width and the maximum connected order of a bramble: the minimum size of a connected vertex set meeting every element of the bramble. We disprove their conjecture by giving an infinite family of counterexamples in Section 7.…”

Gromov Hyperbolicity in Mycielskian Graphs

“…(see, for example, [6][7][8][9]14,[18][19][20][21][25][26][27][28][29][30][31][32][33][34][35]). x, y ∈ J(G) \ V(G), z ∈ J(G), …”