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.