Related papers: On Graphs Representable by Pattern-Avoiding Words
The representation number of a graph is the minimum number of copies of each vertex required to represent the graph as a word, such that the letters corresponding to vertices $x$ and $y$ alternate if and only if $xy$ is an edge in the…
A graph is called $k$-representable if there exists a word $w$ over the nodes of the graph, each node occurring exactly $k$ times, such that there is an edge between two nodes $x,y$ if and only after removing all letters distinct from…
The graphs with permutation-representation number (\textit{prn}) at most two are known. While a characterization for the class of graphs with the \textit{prn} at most three is an open problem, we summarize the graphs of this class that are…
Representing words by vectors, or embeddings, enables computational reasoning and is foundational to automating natural language tasks. For example, if word embeddings of similar words contain similar values, word similarity can be readily…
We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern…
It has become a de-facto standard to represent words as elements of a vector space (word2vec, GloVe). While this approach is convenient, it is unnatural for language: words form a graph with a latent hierarchical structure, and this…
Words in some natural languages can have a composite structure. Elements of this structure include the root (that could also be composite), prefixes and suffixes with which various nuances and relations to other words can be expressed.…
Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism…
The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping…
Attributes of words and relations between two words are central to numerous tasks in Artificial Intelligence such as knowledge representation, similarity measurement, and analogy detection. Often when two words share one or more attributes…
A word-graph Gw is a digraph represented by a word w such that the vertex-set V(Gw) is the alphabet of w and the edge-set E(Gw) is determined by non-identical adjacent letter pairs in w. In this paper we study the strong-connectivity of…
Recent work at the intersection of formal language theory and graph theory has explored graph grammars for graph modeling. However, existing models and formalisms can only operate on homogeneous (i.e., untyped or unattributed) graphs. We…
The representation of graphs is commonly based on the adjacency matrix concept. This formulation is the foundation of most algebraic and computational approaches to graph processing. The advent of deep learning language models offers a wide…
Graph-based text representation focuses on how text documents are represented as graphs for exploiting dependency information between tokens and documents within a corpus. Despite the increasing interest in graph representation learning,…
The notion of a $p$-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a $p$-Riordan word, and show how to encode…
The notion of $12$-representable graphs was introduced as a variant of a well-known class of word-representable graphs. Recently, these graphs were shown to be equivalent to the complements of simple-triangle graphs. This indicates that a…
Words can be represented by composing the representations of subword units such as word segments, characters, and/or character n-grams. While such representations are effective and may capture the morphological regularities of words, they…
The vast amount of data and increase of computational capacity have allowed the analysis of texts from several perspectives, including the representation of texts as complex networks. Nodes of the network represent the words, and edges…
A graph $G = (V, E)$ is said to be word-representable if a word $w$ can be formed using the letters of the alphabet $V$ such that for every pair of vertices $x$ and $y$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. Gaetz and Ji…
In this short note, we first associate a new simple undirected graph with a given word over an ordered alphabet of $n$-letters. We will call it the Lyndon graph of that word. Then, we introduce the concept of the Lyndon-word representable…