Representing Graphs via Pattern Avoiding Words

The notion of a word-representable graph has been studied in a series of papers in the literature. 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 $xy$ is an edge in $E$. If $V =\{1, \ldots, n\}$, this is equivalent to saying that $G$ is word-representable if for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if the subword $w_{\{x,y\}}$ of $w$ consisting of all occurrences of $x$ or $y$ in $w$ has no consecutive occurrence of the pattern 11. In this paper, we introduce the study of $u$-representable graphs for any word $u \in \{1,2\}^*$. A graph $G$ is $u$-representable if and only if there is a labeled version of $G$, $G=(\{1, \ldots, n\}, E)$, and a word $w \in \{1, \ldots, n\}^*$ such that for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if $w_{\{x,y\}}$ has no consecutive occurrence of the pattern $u$. Thus, word-representable graphs are just $11$-representable graphs. We show that for any $k \geq 3$, every finite graph $G$ is $1^k$-representable. This contrasts with the fact that not all graphs are 11-representable graphs. The main focus of the paper is the study of $12$-representable graphs. In particular, we classify the $12$-representable trees. We show that any $12$-representable graph is a comparability graph and the class of $12$-representable graphs include the classes of co-interval graphs and permutation graphs. We also state a number of facts on $12$-representation of induced subgraphs of a grid graph

On the 12-representability of induced subgraphs of a grid graph

The notion of a 12-representable graph was introduced by Jones, Kitaev, Pyatkin and Remmel in [Representing graphs via pattern avoiding words, Electron. J. Combin. 22 (2015) #P2.53]. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph, and also that a tree is 12-representable if and only if it is a double caterpillar. Moreover, Jones et al. initiated the study of 12- representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question of Jones et al. is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs. In this paper we answer the question of Jones et al. by providing a complete characterization of 12-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area

Графы, представимые в виде слов : обзор результатов

Letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word xyxy · · · (of even or odd length) or a word yxyx · · · (of even or odd length). A graph G = (V,E) is word-representable if and only if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy ∈ E. Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. This paper is a comprehensive survey on the theory of word-representable graphs and it includes the most recent developments in the area

The combinatorics of Jeff Remmel

We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics