Suchanda Roy

Suchanda Roy, Ramesh Hariharasubramanian

A graph $G=(V,E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that two distinct letters $x,y\in V$ alternate in $w$ if and only if $xy \in E$. Word-representable graphs form a well-studied graph class with connections to graph orientations, combinatorics on words, and graph coloring. A near-triangulation is a planar graph in which every face except the outer face is a triangle. Several subclasses of near-triangulations have previously been investigated in the context of word-representability, including polyomino triangulations, triangulations of rectangular polyominoes with a single domino tile, $K_4$-free near-triangulations, face subdivisions of triangular grid graphs, triangulations of grid-covered cylinder graphs, and chordal near-triangulations. In this paper, we obtain a complete characterization of word-representable near-triangulations in terms of forbidden induced subgraphs. Our result unifies and extends the previously known characterizations for the above subclasses, while also correcting inaccuracies appearing in earlier works.

Suchanda Roy, Ramesh Hariharasubramanian

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set consists of all increasing $k$-tuples $(x_1,x_2,\dots,x_k)$ with $1\le x_1<x_2<\cdots<x_k\le n$, where two vertices $(x_1,\dots,x_k)$ and $(y_1,\dots,y_k)$ are adjacent whenever $x_{i+1}=y_i$ for all $1\le i\le k-1$ or $y_{i+1}=x_i$ for all $1\le i\le k-1$. Shift graphs are classical examples of sparse graphs having arbitrarily high chromatic number and odd girth. We further observe that shift graphs arise naturally as induced subgraphs of simplified de Bruijn graphs. Although simplified de Bruijn graphs contain non-word-representable members in general, we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable. As a consequence, we obtain an explicit family of graphs exhibiting a contrast between line graph and line digraph constructions: there exists a family of word-representable graphs whose line graphs are not word-representable when the number of vertices is at least $5$, while their line digraphs are word-representable.