Graph Operations Characterizing Rank-Width and Balanced Graph Expressions

Abstract

Graph complexity measures like tree-width, clique-width, NLC-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. Rank-width is based on ranks of adjacency matrices of graphs over GF(2). We propose here algebraic operations on graphs that characterize rank-width. For algorithmic purposes, it is important to represent graphs by balanced terms. We give a unique theorem that generalizes several "balancing theorems" for tree-width and clique-width. New results are obtained for rank-width and a variant of clique-width, called m-clique-width.

Better Polynomial Algorithms on Graphs of Bounded Rank-Width

“…In a search for a "more suitable form" of a rank-decomposition, Courcelle and Kanté [4] defined the bilinear products of multiple-coloured graphs, and proposed algebraic expressions over these operators as an equivalent description of a rankdecomposition (cf. Theorem 3.1).…”

“…. , t} is the set of labels (this notion is exactly equivalent to multiplecoloured graphs of [4]). Having a graph G with an (implicitly) associated tlabeling lab, we refer to the pair (G, lab) as to a t-labeled graph and use notationḠ.…”

“…So even an unlabeled graph can be considered as t-labeled with no labels, and every t-labeled graph is also t -labeled for all t > t. We will often view a t-labeling of G equivalently as a mapping V (G) → GF(2) t to the binary vector space of dimension t (cf. [4] …”

“…be a binary operator, called t-labeling composition (as bilinear product of [4]), over pairs of t-labeled graphs…”

“…The core of the new approach lies in an alternative characterization of rankdecompositions using bilinear graph products [4], or equivalently labeling parse trees [10,11], or R k -joins [2]. These new approaches have led to interesting new FPT algorithms [2,12] which run much faster with respect to the rank-width parameter than previously known algorithms, as in [5].…”