A hole is a chordless cycle with at least 4 vertices, and is even if it has an even number of vertices. A cap consists of a cycle with at least 5 vertices with exactly one chord and that chord creates a triangle with the cycle. A graph is (cap, even hole)-free if it has no induced cap or even hole.

We give an explicit construction of (cap, even-hole)-free graphs. We prove that (triangle, even hole)-free graphs have treewidth at most 5 and that (cap, even hole)-free graphs without clique cutsets have treewidth at most $6\omega(G)-1$, and that both of these classes have clique-width at most 48. We use these results to give an $O(nm)$ algorithm for $q$-colouring (cap, even-hole)-free graphs for fixed $q$ and an $O(nm)$ algorithm for maximum weight stable set, where, as usual, $n$ is the number of vertices and $m$ the number of edges of the input graph. We also give a polynomial-time algorithm for minimum colouring.

This is joint work with Murilo V. G. da Silva, Shenwei Huang and Kristina Vu\v{s}kovi\'c.