6 research outputs found

    Representing Graphs via Pattern Avoiding Words

    Get PDF
    The notion of a word-representable graph has been studied in a series of papers in the literature. A graph G=(V,E)G=(V,E) is word-representable if there exists a word ww over the alphabet VV such that letters xx and yy alternate in ww if and only if xyxy is an edge in EE. If V={1,,n}V =\{1, \ldots, n\}, this is equivalent to saying that GG is word-representable if for all x,y{1,,n}x,y \in \{1, \ldots, n\}, xyExy \in E if and only if the subword w{x,y}w_{\{x,y\}} of ww consisting of all occurrences of xx or yy in ww has no consecutive occurrence of the pattern 11. In this paper, we introduce the study of uu-representable graphs for any word u{1,2}u \in \{1,2\}^*. A graph GG is uu-representable if and only if there is a labeled version of GG, G=({1,,n},E)G=(\{1, \ldots, n\}, E), and a word w{1,,n}w \in \{1, \ldots, n\}^* such that for all x,y{1,,n}x,y \in \{1, \ldots, n\}, xyExy \in E if and only if w{x,y}w_{\{x,y\}} has no consecutive occurrence of the pattern uu. Thus, word-representable graphs are just 1111-representable graphs. We show that for any k3k \geq 3, every finite graph GG is 1k1^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 1212-representable graphs. In particular, we classify the 1212-representable trees. We show that any 1212-representable graph is a comparability graph and the class of 1212-representable graphs include the classes of co-interval graphs and permutation graphs. We also state a number of facts on 1212-representation of induced subgraphs of a grid graph

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

    Get PDF
    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

    Get PDF
    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

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

    Representing Graphs via Pattern-Avoiding Words

    No full text
    corecore