A simple linear-time algorithm for finding path-decompositions of small width

A connected path decomposition of a simple graph G is a path decomposition ( X 1 , … , X l ) such that the subgraph of G induced by X 1 ∪ ⋯ ∪ X i is connected for each i ∈ { 1 , … , l }. The connected pathwidth of G is then the minimum ...

In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that ...

We prove that every graph of rank-width k is a pivot-minor of a graph of tree-width at most 2k. We also prove that graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs of linear rank-...