Minimum length word-representants of word-representable graphs

Abstract

A graph G=(V,E) is said to be word-representable if a word w can be formed using the letters of the alphabet V such that for every pair of vertices x and y, xy∈E if and only if x and y alternate in w. Gaetz and Ji have recently introduced the notion of minimum length word-representants for word-representable graphs. They have also determined the minimum possible length of the word-representants for certain classes of graphs, such as trees and cycles.In this paper, we solved two of the three cases and partially solved the remaining case of an open problem presented in Gaetz and Ji (2020) by Gaetz and Ji. This problem asks to classify all 2-word-representable graphs based on the length of their minimum length word-representants. Furthermore, we found a relation between the maximum and minimum number of occurrences of letters in minimum length word-representants and the diameter of the graph. Finally, we gave an upper bound on the minimum possible length of a word-representant of a graph under certain word-representability preserving graph operations such as connecting two graphs by an edge and gluing two graphs in a vertex.