In this paper, we study the limit of compactness which is a graph index originally

introduced for measuring structural characteristics of hypermedia. Applying

compactness to large scale small-world graphs [1] observed its limit behaviour to be

equal 1. The striking question concerning this finding was whether this limit behaviour

resulted from the specifics of small-world graphs or was simply an artefact. In this

paper, we determine the necessary and sufficient conditions for any sequence of

connected graphs resulting in a limit value of C_B = 1 which can be generalized with

some consideration for the case of disconnected graph classes (Theorem 3). This result

can be applied to many well-known classes of connected graphs. Here, we illustrate it by

considering four examples. In fact, our proof-theoretical approach allows for quickly

obtaining the limit value of compactness for many graph classes sparing computational

costs.