Let $G=(V,E)$ be a finite undirected graph. Orient the edges of G in an arbitrary way. A 2-cycle on G is a function $d:E^2\to \mathbb{Z}$ such for each edge e, $d(e,\cdot )$ and $d(\cdot ,e)$ are circulations on G, and $d(e,f)=0$ whenever e and f have a common vertex. We show that each 2-cycle is a sum of three special types of 2-cycles: cycle-pair 2-cycles, Kuratowski 2-cycles, and quad 2-cycles. In the case that the graph is Kuratowski connected, we show that each 2-cycle is a sum of cycle-pair 2-cycles and at most one Kuratowski 2-cycle. Furthermore, if the graph is Kuratowski connected, we characterize when every Kuratowski 2-cycle is a sum of cycle-pair 2-cycles. A consequence of this is that if G is Kuratowski connected and either G is planar or G does not have a linkless embedding, then each 2-cycle on G is a sum of cycle-pair 2-cycles. A 2-cycle d on G is skew-symmetric if $d(e,f)=-d(f,e)$ for all edges $e,f\in E$. We show that each skew-symmetric 2-cycle is a sum of two special types of skew-symmetric 2-cycles: skew-symmetric cycle-pair 2-cycles and skew-symmetric quad 2-cycles. In the case that the graph is Kuratowski connected, we show that each skew-symmetric 2-cycle is a sum of skew-symmetric cycle-pair 2-cycles. Similar results like this had previously been obtained by one of the authors for symmetric 2-cycles. Symmetric 2-cycles are 2-cycles d such that $d(e,f)=d(f,e)$ for all edges $e,f\in E$.

让$G=（V,乙）$是有限无向图。以任意方式定向G的边。G上的 2 周期是一个函数$d：{乙}^2\to \mathbb{Z}$这样对于每条边e，$d（e,\cdot ）$和$d（\cdot ,e）$是G上的循环，并且$d（e,F）=0$每当e和f有公共顶点时。我们证明每个 2 周期是三种特殊类型 2 周期的总和：周期对 2 周期、Kuratowski 2 周期和四 2 周期。在图是 Kuratowski 连接的情况下，我们表明每个 2-cycle 是循环对 2-cycle 的总和，并且最多有一个 Kuratowski 2-cycle。此外，如果图是 Kuratowski 连接的，我们可以表征每个 Kuratowski 2 周期是周期对 2 周期之和。这样做的结果是，如果G是 Kuratowski 连接的并且G是平面的或者G没有无链接嵌入，则G上的每个 2 周期都是周期对 2 周期的总和。G上的2 周期d是斜对称的，如果$d（e,F）=-d（F,e）$对于所有边缘$e,F\epsilon 乙$。我们证明每个斜对称 2 周期是两种特殊类型的斜对称 2 周期的总和：斜对称周期对 2 周期和斜对称四 2 周期。在图是 Kuratowski 连接的情况下，我们证明每个斜对称 2 循环都是斜对称循环对 2 循环之和。一位作者之前曾针对对称 2 周期获得过类似的结果。对称 2 周期是 2 周期d，使得$d（e,F）=d（F,e）$对于所有边缘$e,F\epsilon 乙$。