Related papers: On the Representation Number of Bipartite Graphs

A graph $G = (V, E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct letters $x, y \in V$, the letters $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is…

The problems of determining the permutation-representation number (prn) and the representation number of bipartite graphs are open in the literature. Moreover, the decision problem corresponding to the determination of the prn of a…

A graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is said to be word-representable if there exists a word $w$ over the alphabet $V(G)$ such that, for any two distinct letters $x,y \in V(G)$, the letters $x$ and $y$ alternate in $w$ if…

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective…

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. The word-representability of split graphs was studied in a series of papers in the literature, and the class of word-representable split…

A graph $G = (V, E)$ is word-representable, if there exists a word $w$ over the alphabet $V$ such that for letters $\{x,y\}\in V$, $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is co-bipartite if its complement is a…

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 $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several…

In this work, we show that the class of word-representable graphs is closed under split recomposition and determine the representation number of the graph obtained by recomposing two word-representable graphs. Accordingly, we show that the…

A graph $G=(V,E)$ is representable if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. If $W$ is $k$-uniform (each letter of $W$ occurs exactly $k$…

The notion of word-representable graphs is a generalization of comparability graphs, in which graphs are represented by words. The complexity of word-representation of a word-representable graph is captured through the representation…

While the problem of determining the representation number of an arbitrary word-representable graph is NP-hard, this problem is open even for bipartite graphs. The representation numbers are known for certain bipartite graphs including all…

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 $y$, $x\neq y$, alternate in $w$ if and only if $xy\in E$. A split graph is a graph in which the vertices can be…

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 $E$. It is known that any word-representable graph $G$ is…

Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the…

A simple 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 $w$ iff $xy\in E$. Word-representable graphs generalize several important classes of graphs. A graph…

Word-representable graphs are a subset of graphs that may be represented by a word $w$ over an alphabet composed of the vertices in the graph. In such graphs, an edge exists if and only if the occurrences of the corresponding vertices…

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…

In this work, we characterize the class of word-representable graphs with respect to the modular decomposition. Consequently, we determine the representation number of a word-representable graph in terms of the permutation-representation…

The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our…

For an arbitrary word $w$ on an alphabet, we can define the alternating symbol graph, $G(w)$, as the graph in which the edge $(a, b)$ is in $E$ iff the letters $a$ and $b$ alternate in the word $w$. A graph $G = (V, E)$ is said to be…