The notion of word-representable graphs is a generalization of comparability graphs, in which graphs are represented by words. The complexity of word-representation of a word-representable graph is captured through the representation number, whereas the corresponding concept is the permutation-representation number for comparability graphs. The graphs with the (permutation-)representation number at most two were characterized in the literature. While certain examples in the class of graphs with the (permutation-)representation number three are known, no characterization for these classes is available. In this work, we prove that the representation number of melon graphs is at most three. Further, we characterize the class of melon graphs restricted to comparability graphs and show that their permutation-representation number is also at most three. Moreover, this work characterizes the word-representable line graphs of melon graphs and establishes that their representation number is at most three.

翻译：词可表示图的概念是可比较图的一种推广，其中图由词表示。词可表示图的表示复杂度通过表示数来刻画，而可比较图对应的概念则是置换表示数。表示数（置换表示数）不超过二的图已在文献中得到刻画。虽然已知表示数（置换表示数）为三的图类中存在某些实例，但这些图类尚未有完整的刻画。在本工作中，我们证明了甜瓜图的表示数不超过三。进一步，我们刻画了限制为可比较图的甜瓜图类，并证明其置换表示数也不超过三。此外，本工作刻画了甜瓜图的词可表示线图，并确定其表示数不超过三。