Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (e.g. the class of perfect graphs and the class of even-hole-free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. Furthermore, we obtain polynomial χ-bounding functions for these classes.

Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (eg, the class of perfect graphs and the class of even‐hole‐free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. Furthermore, we obtain polynomial χ‐bounding functions for these classes.

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In this paper, we study the class of graphs scriptC defined by excluding the following structures as induced subgraphs: theta, pyramid, 1‐wheel, and 3‐wheel. We describe the structure of graphs in scriptC, and we give a polynomial‐time recognition algorithm for this class. We also prove that K 4‐free graphs in scriptC are 4‐colorable. We remark that scriptC includes the class of chordal graphs, as well as the class of line graphs of triangle‐free graphs.