Decomposition for a certain class of graphs

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?

• Did you check the condition that each graph vertex appears in a connected subtree of bags of the decomposition? – David Eppstein Jan 7 at 6:35
• 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 – Student Jan 7 at 6:58