# (Theta, triangle)-free and (even hole, K_4)-free graphs. Part 2 : bounds on treewidth

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_1b_2b_3 is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to a. An even hole is a chordless cycle of even length. For three non-negative integers i≤ j≤ k, let S_i,j,k be 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.

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