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Partitioning a graph into disjoint cliques and a triangle-free graph

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Figure

Figure 2: red negators: the sun component (left) and the bull-free component(right)
Figure 5: The literal gadget (left) with endpoints x, y, z and the propagatorgadget (right) with endpoints u, v, w along with a partition where the whitevertices are in the P3-free part and the black vertices are in K3-free part
Figure 6: The graphs B1, B2, B3, B4, B5 and B6
Figure 7: The graphs J1, J2, J3 and J4
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