Enumeration and Extensions of Word-Representants

Abstract

Given a finite word w over a finite alphabet V, we may construct a graph with vertex set V and an edge between to elements of V if and only if they alternate in the word w. This is the notion of word-representability of graphs. In this paper, we first study minimal length words which represent graphs, giving an explicit formula for both the length and the number of such words in the case of trees and cycles. Then we extend this notion to study the graphs representable with other patterns in words, proving in all cases aside from one (still unknown to us), all graphs are representable by all other patterns. Finally, we pose a few open problems for further work.