Fuzzy line graphs extend the classical line-graph construction so that edge adjacency is recorded as a graded membership value rather than a binary fact, and the resulting construction, together with its basic transformation and connectivity theory, was developed by Mohammad [1]. That framework, however, records only a single degree of confidence for each link and has no place for an explicit degree of doubt. Many of the systems it is meant to model—unreliable sensor links, contested survey data, roads whose condition reports conflict are more naturally described by an intuitionistic fuzzy graph, in which every vertex and edge carries both a membership degree and an independent non-membership degree, leaving a hesitation margin between them. This paper extends the fuzzy line graph construction of [1] to this intuitionistic setting. We define the intuitionistic fuzzy line graph (IFLG), prove that the construction is well defined and order-preserving in both the membership and the non-membership component, and prove that fuzzy connectivity is preserved under the extended construction. We then prove a Hesitation Interpolation Lemma, a result with no counterpart in the purely fuzzy case, showing that the hesitation degree of every edge produced by the construction lies between the hesitation degrees of the two edges it joins, so that uncertainty about a link is neither manufactured nor erased by the transformation. The framework is illustrated on a worked example and on a transportation network in which each road is assigned both a confidence and a doubt value, and the discussion compares the intuitionistic construction with its fuzzy predecessor, showing the fuzzy line graph to be the special case in which hesitation is uniformly zero.