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Characterization of the Congestion Lemma on Layout Computation

An embedding of a guest network 
 
 G
 N
 
 into a host network 
 
 H
 N
 
 is to find a suitable bijective function between the vertices of the guest and the host such that each link of 
 
 G
 N
 
 is stretched to a path in 
 
 H
 N
 
 . The layout measure is attained by counting the length of paths in 
 
 H
 N
 
 corresponding to the links in 
 
 G
 N
 
 and with a complexity of finding the best possible function overall graph embedding. This measure can be computed by summing the minimum congestions on each link of 
 
 H
 N
 
 , called the congestion lemma. In the current study, we discuss and characterize the congestion lemma by considering the regularity and optimality of the guest network. The exact values of the layout are generally hard to find and were known for very restricted combinations of guest and host networks. In this series, we derive the correct layout measures of circulant networks by embedding them into the path- and cycle-of-complete graphs.