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Connectivity is one of the important fundamental structural properties of graphs, and a graph drawing D should faithfully represent the connectivity structure of the underlying graph G. This paper investigates connectivity-faithful graph drawing leveraging the famous Nagamochi-Ibaraki (NI) algorithm, which computes a sparsification G_NI, preserving the k-connectivity of a k-connected graph G. Specifically, we first present CFNI, a divide-and-conquer algorithm, which computes a sparsification G_CFNI, which preserves the global k-connectivity of a graph G and the local h-connectivity of the h-connected components of G. We then present CFGD, a connectivity-faithful graph drawing algorithm based on CFNI, which faithfully displays the global and local connectivity structure of G. Extensive experiments demonstrate that CFNI outperforms NI with 66% improvement in the connectivity-related sampling quality metrics and 73% improvement in proxy quality metrics. Consequently, CFGD outperforms a naive application of NI for graph drawing, in particular with 62% improvement in stress metrics. Moreover, CFGD runs 51% faster than drawing the whole graph G, with a similar quality.