Clique-width is a complexity measure of directed as well as undirected graphs. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to edge-colored graphs. A C-colored graph is a graph where the arcs are colored with colors from the set C. There is not a natural notion of rank-width for C-colored graphs. We define two notions of rank-width for them, both based on a coding of C-colored graphs by edge-colored graphs where each edge has exactly one color from a field F and named respectively F-rank-width and F-bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for F-colored graphs and prove that F-colored graphs of bounded F-rank-width are characterised by a finite list of F-colored graphs to exclude as vertex-minors. A cubic-time algorithm to decide whether a F-colored graph has F-rank-width (resp. F-bi-rank-width) at most k, for fixed k, is also given. Graph operations to check MSOL-definable properties on F-colored graphs of bounded rank-width are presented. A specialisation of all these notions to (directed) graphs without edge colors is presented, which shows that our results generalise the ones in undirected graphs.
本文证明了一个连通图的线性秩宽为1当且仅当它是一个距离遗传图且它的分裂分解树是一条路.一个直接的结果是，我们可以在线性时间内判定一个图的线性秩宽是否至多为1，如果不是，则给出一个障碍.其他直接的结果是线性秩宽为1的图的几个特征。特别地，一个连通图具有线性秩宽1当且仅当它局部等价于一个毛毛虫当且仅当它是一条路的顶点子图[O-juanKwon and Sang-il Oum，Graphs of small rank-width are approximates-minor of graphs of small tree-width，arxiv：1203.3606]当且仅当它不包含co-K_2图、Net图和5-圈图作为顶点子图[Isolde Adler，Arthur M. Farley和Andrzej Proskurowski，线性秩宽度至多为1的障碍，arxiv：1106.2533]。
Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs
在[Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices，arXiv：1007.3807v1]中，Oum证明了，对于一个固定的有限域$\mathbf{F}$，在有界$\mathbf{F}$ -rank-width的$\mathbf{F}$上的（反）对称矩阵的任何无限序列$M_1,M_2,...$都有一对$i< j$，使得$M_i$同构于$M_j$的主枢轴变换的主子矩阵。我们将这个结果推广到Rao和我在[边着色图的秩宽度，arXiv：0709.1433v4]中介绍的$σ$对称矩阵。（斜）对称矩阵是PDMATHAMR-对称矩阵的特殊情况。作为一个副产品，我们得到，对于每一个无限序列$G_1,G_2,...$的有界秩宽度的有向图，存在一对$i<j$，使得$G_i$是一个非子式$G_j$。另一个推论是，PDMATHAMR对称矩阵的非奇异主子矩阵形成δ拟阵。我们以这种方式扩展了Bouchet的Delta拟阵可表示性的概念。
Branch-width of connectivity functions is fixed-parameter tractable
University of Copenhagen（哥本哈根大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）；Department of Mathematical Sciences, KAIST（韩国科学技术院数理科学系）
