Isolde Adler

Directed tree-width examples

In [T. Johnson, N. Robertson, P.D. Seymour, R. Thomas, Directed tree-width, J. Combin. Theory Ser. B 82 (2001) 138-154] Johnson, Robertson, Seymour and Thomas define the notion of directed tree-width, dtw(D), of a directed graph D. They ask whether dtw(D)>=k-1 implies that D has a haven of order k. A negative answer is given. Furthermore they define a generalisation of the robber and cops game of [P.D. Seymour, R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993) 22-33] to digraphs. They ask whether it is true that if k cops can catch the robber on a digraph, then they can do so robber-monotonely. Again a negative answer is given. We also show that contraction of butterfly edges can increase directed tree-width.

Linear rank-width and linear clique-width of trees

We show that for every forest T the linear rank-width of T is equal to the path-width of T, and the linear clique-width of T equals the path-width of T plus two, provided that T contains a path of length three. It follows that both linear rank-width and linear clique-width of forests can be computed in linear time. Using our characterization of linear rank-width of forests, we determine the set of minimal excluded acyclic vertex-minors for the class of graphs of linear rank-width at most k.

Interpreting nowhere dense graph classes as a classical notion of model theory

A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of complete graphs that occur as r-minors. We observe that this recent tameness notion from (algorithmic) graph theory is essentially the earlier stability theoretic notion of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence property. Expressed in terms of PAC learning, the concept classes definable in first-order logic in a subgraph-closed graph class have bounded sample complexity, if and only if the class is nowhere dense.

Obstructions for linear rank-width at most 1

We establish the set of minimal forbidden induced subgraphs for the class of graphs having linear rank-width at most 1. From these we derive both the vertex-minor and pivot-minor obstructions for the class.

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank-width at most 1, can be computed in time

Linear Rank-Width of Distance-Hereditary Graphs

{\mathcal {O}}(n^2\cdot \log _2 n)

Fast minor testing in planar graphs

O(n2·log2n), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every n-element matroid of branch-width at most 2 can be computed in time