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]。
英文摘要：
We prove that a connected graph has linear rank-width 1 if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear rank-width at most 1, and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rank-width 1. In particular a connected graph has linear rank-width 1 if and only if it is locally equivalent to a caterpillar if and only if it is a vertex-minor of a path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors of graphs of small tree-width, arxiv:1203.3606] if and only if it does not contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors [Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for linear rank-width at most 1, arxiv:1106.2533].
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）
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。
