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A  theta  is a graph made of three internally vertex‐disjoint chordless paths    P  1   =  a  …  b  ,   P  2   =  a  …  b  ,    P  3   =  a  …  b   of length at least 2 and such that no edges exist between the paths except the three edges incident to   a   and the three edges incident to   b  . A  pyramid  is a graph made of three chordless paths    P  1   =  a  …   b  1   ,   P  2   =  a  …   b  2   ,    P  3   =  a  …   b  3    of length at least 1, two of which have length at least 2, vertex‐disjoint except at   a  , and such that    b  1    b  2    b  3    is a triangle and no edges exist between the paths except those of the triangles and the three edges incident to   a  . An  even hole  is a chordless cycle of even length. For three nonnegative integers   i  ≤  j  ≤  k  , let    S   i  ,  j  ,  kbe the tree with a vertex   v  , from which start three paths with   i  ,  j  , and   k   edges, respectively. We denote by    K  t    the complete graph on   t   vertices. We prove that for all nonnegative integers   i  ,  j  ,  k  , the class of graphs that contain no theta, no    K  3   , and no    S   i  ,  j  ,  kas induced subgraphs have bounded treewidth. We prove that for all nonnegative integers   i  ,  j  ,  k  ,  t  , the class of graphs that contain no even hole, no pyramid, no    K  t   , and no    S   i  ,  j  ,  kas induced subgraphs have bounded treewidth. To bound the treewidth, we prove that every graph of large treewidth must contain a large clique or a minimal separator of large cardinality.

(Theta, triangle)‐free and (even hole, K 4 )‐free graphs. Part 2: Bounds on treewidth