### Abstract

Original language | English |
---|---|

Article number | P2.53 |

Number of pages | 20 |

Journal | The Electronic Journal of Combinatorics |

Volume | 22 |

Issue number | 2 |

Early online date | 15 Jun 2015 |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- word-representable graphs
- ladder graphs
- grid graphs
- permutation graphs
- co-interval graphs
- comparability graphs
- pattern avoidance

### Cite this

*The Electronic Journal of Combinatorics*,

*22*(2), [P2.53].

}

*The Electronic Journal of Combinatorics*, vol. 22, no. 2, P2.53.

**Representing graphs via pattern avoiding words.** / Jones, Miles; Kitaev, Sergey; Pyatkin, Artem; Remmel, Jeffrey.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Representing graphs via pattern avoiding words

AU - Jones, Miles

AU - Kitaev, Sergey

AU - Pyatkin, Artem

AU - Remmel, Jeffrey

PY - 2015

Y1 - 2015

N2 - 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,...,n}, this is equivalent to saying that G is word-representable if for all x,y ϵ {1,…,n}, xy ϵ 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 ϵ {1, 2}*. A graph G is u -representable if and only if there is a vertex-labeled version of G, G = ( {1,…,n},E ), and a word w ϵ {1,…,n}* such that for all x,y ϵ {1,…,n}, xy ϵ 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 > 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.

AB - 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,...,n}, this is equivalent to saying that G is word-representable if for all x,y ϵ {1,…,n}, xy ϵ 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 ϵ {1, 2}*. A graph G is u -representable if and only if there is a vertex-labeled version of G, G = ( {1,…,n},E ), and a word w ϵ {1,…,n}* such that for all x,y ϵ {1,…,n}, xy ϵ 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 > 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.

KW - word-representable graphs

KW - ladder graphs

KW - grid graphs

KW - permutation graphs

KW - co-interval graphs

KW - comparability graphs

KW - pattern avoidance

UR - http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p53

M3 - Article

VL - 22

JO - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

M1 - P2.53

ER -