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• A directed convolutional neural network model is proposed • Partial dilation-Global erosion generates resistance against information loss • Several recent approaches for the segmentation of graphemes/characters are discussed • The over-traces issue is addressed for the first time in the literature • MMOs can provide real solutions if a convenient strategy is adopted Due to the nature of Arabic handwriting, segmenting words into characters/graphemes is the most difficult and critical task of the recognition system. The present paper proposes an approach to segment handwritten Arabic words into graphemes based on a directed Convolutional Neural Network (CNN) and Mathematical Morphology Operations (MMO). Arabic script is cursive, which means that almost all graphemes are connected via horizontal links; therefore, a technique to remove links will facilitate the segmentation of graphemes. In general, an MMO such as erosion seems suitable for getting the job done, but since Arabic handwriting is difficult, MMOs cause information loss and suffer from many issues such as diacritics and over-traces, which lead to over/under/bad segmentations. To overcome limitations, the present paper addresses these issues in the following order: the over-traces issue is addressed for the first time in the literature; a robust algorithm for diacritics extraction is provided; and finally, the main segmentation algorithm adopts a strategy based on a Partial Dilation (PD)-Global Erosion (GE) technique to combat the information loss issue. The PD phase amplifies important regions, while GE eliminates links between graphemes. The complementarity between PD and GE facilitates the extraction of graphemes and creates resistance against information loss. To properly tackle these difficult problems, this article exploits the robustness of CNNs, so a new directed CNN model is suggested. The idea is to draw the model's attention to certain targeted features, which are selected according to the nature of the problem addressed. The proposed directed CNN is used in all phases of the segmentation process. The experimental results are very encouraging and show that the proposed directed CNN model outperformed basic CNN in many experiments. The results also reveal that the followed strategy improved the ability of MMOs to perform segmentation and to compete with other approaches in this research area. [ABSTRACT FROM AUTHOR]