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

HH-product and HH-threshold graphs

This paper is the continuation of the research of the author and his colleagues of the {\it canonical} decomposition of graphs. The idea of the canonical decomposition is to define the binary operation on the set of graphs and to represent the graph under study as a product of prime elements with respect to this operation. We consider the graph together with the arbitrary partition of its vertex set into $n$ subsets ($n$-partitioned graph). On the set of $n$-partitioned graphs distinguished up to isomorphism we consider the binary algebraic operation $\circ_H$ ($H$-product of graphs), determined by the digraph $H$. It is proved, that every operation $\circ_H$ defines the unique factorization as a product of prime factors. We define $H$-threshold graphs as graphs, which could be represented as the product $\circ_{H}$ of one-vertex factors, and the threshold-width of the graph $G$ as the minimum size of $H$ such, that $G$ is $H$-threshold. $H$-threshold graphs generalize the classes of threshold graphs and difference graphs and extend their properties. We show, that the threshold-width is defined for all graphs, and give the characterization of graphs with fixed threshold-width. We study in detail the graphs with threshold-widths 1 and 2

Linear Rank-Width of Distance-Hereditary Graphs

We present a characterization of the linear rank-width of distance-hereditary graphs. Using the characterization, we show that the linear rank-width of every n-vertex distance-hereditary graph can be computed in time O(n²⋅log(n)), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. For our characterization, we combine modifications of canonical split decompositions with an idea of [Megiddo, Hakimi, Garey, Johnson, Papadimitriou: The complexity of searching a graph. JACM 1988], used for computing the path-width of trees. We also provide a set of distance-hereditary graphs which contains the set of distance-hereditary vertex-minor obstructions for linear rank-width. The set given in [Jeong, Kwon, Oum: Excluded vertex-minors for graphs of linear rank-width at most k. STACS 2013: 221–232] is a subset of our obstruction set

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM