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 vertex that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins, and consequently obtain a polynomial time recognition algorithm for the class. In this paper we further use this decomposition theorem to obtain polynomial time algorithms for maximum weight clique, maximum weight stable set and coloring problems. We also show that for a graph G in the class, if its maximum clique size is ω, then its chromatic number is bounded by $\max \{\omega ,3\}$, and that the class is 3-clique-colorable.

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。轮是由孔和在孔中至少有3个相邻点的顶点形成的图形。在这一系列论文中，我们研究了不包含theta或wheel作为诱导子图的图的类。在该系列的第二部分中，我们证明了该类的分解定理，该分解定理使用集团割集和2-联接，因此获得了该类的多项式时间识别算法。在本文中，我们进一步使用该分解定理来获得多项式时间算法，以解决最大权重集团，最大权重稳定集和着色问题。我们还表明对于图G在该类中，如果其最大派系大小为ω，则其色数为$\mathrm{最高}\{\omega ，3\}$，并且该类是3色可着色的。