Abstract This paper is the first from a series of papers that establish a common analogue of the strong component and basilica decompositions for bidirected graphs. A bidirected graph is a graph in which a sign ＋ or − is assigned to each end of each edge, and therefore is a common generalization of digraphs and signed graphs. Unlike digraphs, the reachabilities between vertices by directed trails and paths are not equal in general bidirected graphs. In this paper, we set up an analogue of the strong connectivity theory for bidirected graphs regarding directed trails, motivated by degree-bounded factor theory. We define the new concepts of circular connectivity and circular components as generalizations of the strong connectivity and strong components. In our main theorem, we characterize the inner structure of each circular component; we define a certain binary relation between vertices in terms of the circular connectivity and prove that this relation is an equivalence relation. The nontrivial aspect of this structure arises from directed trails starting and ending with the same sign, and is therefore characteristic to bidirected graphs that are not digraphs. This structure can be considered as an analogue of the general Kotzig–Lovasz decomposition, a known canonical decomposition in 1-factor theory. From our main theorem, we also obtain a new result in b -factor theory, namely, a b -factor analogue of the general Kotzig–Lovasz decomposition.

中文翻译：

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一般 Kotzig-Lovász 分解的有符号类似物

摘要 本文是一系列论文中的第一篇，这些论文建立了双向图的强分量和大教堂分解的公共类比。双向图是在每条边的每一端都分配一个符号＋或－的图，因此是有向图和有符号图的共同推广。与有向图不同，有向路径和路径的顶点之间的可达性在一般双向图中是不相等的。在本文中，我们为关于有向路径的双向图建立了强连通性理论的类比，受度有界因子理论的启发。我们将循环连通性和循环分量的新概念定义为强连通性和强分量的概括。在我们的主定理中，我们描述了每个圆形组件的内部结构；我们根据循环连通性定义了顶点之间的某种二元关系，并证明这种关系是等价关系。这种结构的重要方面源于以相同符号开始和结束的有向路径，因此是非有向图的双向图的特征。这种结构可以被认为是一般 Kotzig-Lovasz 分解的类似物，这是 1 因子理论中已知的典型分解。从我们的主要定理，我们还得到了 b 因子理论的新结果，即一般 Kotzig-Lovasz 分解的 ab 因子类似物。这种结构的重要方面源于以相同符号开始和结束的有向路径，因此是非有向图的双向图的特征。这种结构可以被认为是一般 Kotzig-Lovasz 分解的类似物，这是 1 因子理论中已知的典型分解。从我们的主要定理，我们还得到了 b 因子理论的新结果，即一般 Kotzig-Lovasz 分解的 ab 因子类似物。这种结构的重要方面源于以相同符号开始和结束的有向路径，因此是非有向图的双向图的特征。这种结构可以被认为是一般 Kotzig-Lovasz 分解的类似物，这是 1 因子理论中已知的典型分解。从我们的主要定理，我们还得到了 b 因子理论的新结果，即一般 Kotzig-Lovasz 分解的 ab 因子类似物。