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Suppose a graph, $G = (V,E)$ is characterized as a lattice/network of cliques as in the picture below. Does there exist some decomposition principle (i.e. on the right) for $G$, that yields some special structure that may be used to explain efficiencies experienced with what are supposed to be combinatorial hard problems?

enter image description here

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  • $\begingroup$ Did you check the condition that each graph vertex appears in a connected subtree of bags of the decomposition? $\endgroup$ – David Eppstein Jan 7 at 6:35
  • $\begingroup$ Sorry I must've overlooked that property. A more proper question would be, does there exist a decomposition technique where the type of described graph yields some structure similar to what is displayed here: imgur.com/a/lGQrVH3 $\endgroup$ – Student Jan 7 at 6:58

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