Enumeration and extensions of word-representants

Abstract

Given a finite word w over a finite alphabet V, consider the graph with vertex set V and with an edge between two elements of V if and only if the two elements alternate in the word w. Such a graph is said to be word-representable or 11-representable by the word w; this latter terminology arises from the phenomenon that the condition of two elements x and y alternating in a word w is the same as the condition of the subword of w induced by x and y avoiding the pattern 11. In this paper, we first study minimal length words that word-represent graphs, giving an explicit formula for both the length and the number of such words in the case of trees and cycles. We then extend the notion of word-representability (or 11-representability) of graphs to t-representability of graphs, for any pattern t consisting of two distinct letters. We prove that every graph is t-representable for several classes of patterns t. Finally, we pose a few open problems.