Enumeration and Extensions of Word-representants

TL;DR

This paper explores the enumeration of minimal words representing graphs through alternation patterns, extends the concept to t-representability, and proves that most graphs are t-representable for any pattern on two letters.

Contribution

It provides explicit formulas for minimal word lengths for trees and cycles, and generalizes word-representability to t-patterns, showing most graphs are t-representable.

Findings

Explicit formulas for minimal word lengths for trees and cycles

Extension of word-representability to t-patterns

Almost all graphs are t-representable for any pattern on two letters

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 which 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 $t$ on two letters. We prove that every graph is $t$-representable for any pattern $t$ on two letters (except for possibly one class of $t$). Finally, we pose a few open problems for future consideration.