Word-Representation of Melon Graphs

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.