A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced subgraph isomorphic to a theta or a wheel, i.e. the class of (theta, wheel)-free graphs. The decomposition theorem uses clique cutsets and 2-joins. Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graph classes. A 2-join is an edge cutset that appeared in decomposition theorems of several complex classes, such as perfect graphs, even-hole-free graphs and others. In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations (which are much harder to use in algorithms). This is a first example of a decomposition theorem that uses just the combination of clique cutsets and 2-joins. This has several consequences. First, we can easily transform our decomposition theorem into a complete structure theorem for (theta, wheel)-free graphs, i.e. we show how every (theta, wheel)-free graph can be built starting from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are (theta, wheel)-free. Such structure theorems are very rare for hereditary graph classes, only a few examples are known. Secondly, we obtain an $\mathcal{O}(n^{4}m)$-time decomposition based recognition algorithm for (theta, wheel)-free graphs. Finally, in Parts III and IV of this series, we give further applications of our decomposition theorem.

图中的空洞是长度至少为4的无弦循环。theta是由同一对不同顶点之间的三个路径形成的图，因此任何两个路径的并集都会引起空洞。轮是由孔和在该孔中至少有3个邻居的节点组成的图形。在本文中，我们获得了图类的分解定理，其中该图不包含与theta或wheel同构的诱导子图，即无（theta，wheel）图的类。分解定理使用集团割集和2-连接。“集团”割集是顶点割集，在基于分解的算法中确实能很好地工作，但是不幸的是，它的通用性不足以分解更复杂的遗传图类。2-join是一种边缘割集，它出现在几个复杂类的分解定理中，例如完美图形，偶数无孔图等。在这些分解定理中，将2个连接点与顶点割集一起使用，而顶点割集比团割割集更通用，例如星形割集及其概括（在算法中更难使用）。这是分解定理的第一个示例，该分解定理仅使用集团割集和2联接的组合。这有几个后果。首先，我们可以轻松地将分解定理转换为无（θ，车轮）图的完整结构定理，即，我们展示了如何从可以明确构造的基本图开始构建每个无（θ，车轮）图。通过规定的合成操作将它们粘合在一起；并且以这种方式建立的所有图形都没有（θ，车轮）。对于遗传图类，这种结构定理非常少见，仅举几个例子。$\mathcal{Ø}（{ñ}^4米）$（θ，wheel）无图的基于时间分解的识别算法。最后，在本系列的第三部分和第四部分中，我们将进一步应用分解定理。