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We give constructive and combinatorial proofs to decide why certain families of slightly irregular graphs have no planar representation and why certain families have such planar representations. Several non-existence results for infinite families as well as for specific graphs are given. For example, the nonexistence of the graphs with \( n = 11 \) and degree sequence \( (5, 5, 5, \ldots, 4) \) and \( n = 13 \) and degree sequence \( (6, 5, 5, \ldots, 5) \) are shown.