A string set avoids overlaps if it has the property that the prefix of each string in the set does not coincide with the suffix of any string in the set. The problem of avoiding overlaps in strings is extensively investigated since it plays an important role in the context of string matching and coding. The notion of overlap can be extended naturally to two dimensions; two pictures [Formula: see text] and [Formula: see text] have an overlap if one can put one corner of [Formula: see text] on some position in [Formula: see text] in such a way that all symbols in the common positions coincide. A picture with no self-overlaps is called unbordered and it is a generalization in two dimensions of an unbordered (or bifix-free) string.We first investigate the problem of generating all unbordered pictures of fixed size. Then, we study particular sets of unbordered pictures that are non-overlapping each others. We show that the frame of the pictures has a special role in defining these sets. In particular, we prove that there exist kinds of synchronization frames that can be put around the pictures of any set to make it non-overlapping. Finally, we present a construction of non-expandable non-overlapping sets of pictures which starts from some property of frame-completeness.

中文翻译：

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避免重叠的图片集

如果字符串集合具有集合中每个字符串的前缀与集合中任何字符串的后缀不重合的属性，则字符串集合可以避免重叠。避免字符串重叠的问题被广泛研究，因为它在字符串匹配和编码的上下文中起着重要作用。重叠的概念可以自然地扩展到二维；两张图片 [公式：见文本] 和 [公式：见文本] 如果可以将 [公式：见文本] 的一个角放在 [公式：见文本] 中的某个位置，使共同立场重合。没有自重叠的图片称为无边框图片，它是无边框（或无双尾）字符串的二维泛化。我们首先研究生成所有固定大小的无边框图片的问题。然后，我们研究彼此不重叠的特定无边框图片集。我们展示了图片的框架在定义这些集合中具有特殊的作用。特别是，我们证明了存在可以放在任何集合的图片周围以使其不重叠的各种同步帧。最后，我们提出了一种不可扩展的非重叠图片集的构造，它从帧完整性的某些属性开始。