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This paper investigates the robust delay-dependent stability problem of a class of linear uncertain system with interval time-varying delay, and proposes less conservative stability criteria for computing the maximum allowable bound of the delay range. The proposed stability criteria are based on the delay central point technique wherein the delay interval is partitioned into two sub-intervals of equal length, and the time variation of a Lyapunov-Krasovskii functional is considered individually for each of these segments in order to reduce the conservatism of the stability criteria. For robust stability conditions, two categories of system uncertainties, namely, time-varying structured and polytopic-type uncertainties are considered. In deriving the stability conditions in LMI framework, neither model transformations, nor bounding techniques using free-weighting matrix variables are employed for dealing the cross-product terms; instead, they are dealt using tighter integral inequalities. This, in turn, makes the proposed approach computationally more efficient and simple. Subsequently, the results obtained are less conservative in the range of allowable delay bound. The effectiveness of the proposed stability criteria is validated through numerical examples.