Semi-Transitive Orientations and Word-Representable Graphs

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 characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.Comment: arXiv admin note: text overlap with arXiv:0810.031

On word-representability of polyomino triangulations

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 $(x,y)$ is an edge in $E$. Some graphs are word-representable, others are not. It is known that a graph is word-representable if and only if it accepts a so-called semi-transitive orientation. The main result of this paper is showing that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. We demonstrate that this statement is not true for an arbitrary polyomino. We also show that the graph obtained by replacing each $4$-cycle in a polyomino by the complete graph $K_4$ is word-representable. We employ semi-transitive orientations to obtain our results

Encoding graphs by words and morphisms

This thesis is related to encoding graphs by words, where we deal with so called word representation of graphs, relevant to them semi-transitive orientations, and more exotic ways to represent graphs via bijections with certain words and pattern-avoiding permutations. In Chapter 2, we introduce a way to define classes of split graphs via iterations of morphisms and present a number of general results on word-representation of such graphs. A particular result obtained by us that goes beyond the study of split graphs, is a characterization of word-representable graphs in terms of permutations of columns of the adjacency matrices. We also provide a complete classification of word-representable split graphs defined by iteration of morphisms using two 2x2 matrices. In Chapter 3, we study families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving as the limit infinite directed split graphs. For each of such a family we ask the question on whether all graphs in the family are oriented semi-transitively (i.e. are semi-transitive) or a finite iteration k of the morphism produces a non-semi-transitive orientation (which will stay non-semi-transitive for all iterations > k). We fully classify semi-transitive infinite directed split graphs in question. In Chapter 4, we present encoding p-Riordan graphs by p-Riordan words, and encoding Riordan graphs by pattern-avoiding permutations. Also, we encode oriented Riordan graphs by balanced words over the alphabet {0, 1, 2}, and provide, as a bi-product, a proof of a known enumerative result about closed walks in the 3-cube.This thesis is related to encoding graphs by words, where we deal with so called word representation of graphs, relevant to them semi-transitive orientations, and more exotic ways to represent graphs via bijections with certain words and pattern-avoiding permutations. In Chapter 2, we introduce a way to define classes of split graphs via iterations of morphisms and present a number of general results on word-representation of such graphs. A particular result obtained by us that goes beyond the study of split graphs, is a characterization of word-representable graphs in terms of permutations of columns of the adjacency matrices. We also provide a complete classification of word-representable split graphs defined by iteration of morphisms using two 2x2 matrices. In Chapter 3, we study families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving as the limit infinite directed split graphs. For each of such a family we ask the question on whether all graphs in the family are oriented semi-transitively (i.e. are semi-transitive) or a finite iteration k of the morphism produces a non-semi-transitive orientation (which will stay non-semi-transitive for all iterations > k). We fully classify semi-transitive infinite directed split graphs in question. In Chapter 4, we present encoding p-Riordan graphs by p-Riordan words, and encoding Riordan graphs by pattern-avoiding permutations. Also, we encode oriented Riordan graphs by balanced words over the alphabet {0, 1, 2}, and provide, as a bi-product, a proof of a known enumerative result about closed walks in the 3-cube

Solving computational problems in the theory of word-representable graphs

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 ∈ E. Word-representable graphs generalize several important classes of graphs. A graph is word-representable if it admits a semi-transitive orientation. We use semi-transitive orientations to enumerate connected non-word-representable graphs up to the size of 11 vertices, which led to a correction of a published result. Obtaining the enumeration results took 3 CPU years of computation. Also, a graph is word-representable if it is k-representable for some k, that is, if it can be represented using k copies of each letter. The minimum such k for a given graph is called graph's representation number. Our computational results in this paper not only include distribution of k-representable graphs on at most 9 vertices, but also have relevance to a known conjecture on these graphs. In particular, we find a new graph on 9 vertices with high representation number. Also, we prove that a certain graph has highest representation number among all comparability graphs on odd number of vertices. Finally, we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original definition, unlike the equivalence of k-representability and word-representability.Publisher PDFPeer reviewe

On graphs with representation number 3

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 $(x,y)$ is an edge in $E$. A graph is word-representable if and only if it is $k$-word-representable for some $k$, that is, if there exists a word containing $k$ copies of each letter that represents the graph. Also, being $k$-word-representable implies being $(k+1)$-word-representable. The minimum $k$ such that a word-representable graph is $k$-word-representable, is called graph's representation number. Graphs with representation number 1 are complete graphs, while graphs with representation number 2 are circle graphs. The only fact known before this paper on the class of graphs with representation number 3, denoted by $\mathcal{R}_3$, is that the Petersen graph and triangular prism belong to this class. In this paper, we show that any prism belongs to $\mathcal{R}_3$, and that two particular operations of extending graphs preserve the property of being in $\mathcal{R}_3$. Further, we show that $\mathcal{R}_3$ is not included in a class of $c$-colorable graphs for a constant $c$. To this end, we extend three known results related to operations on graphs. We also show that ladder graphs used in the study of prisms are $2$-word-representable, and thus each ladder graph is a circle graph. Finally, we discuss $k$-word-representing comparability graphs via consideration of crown graphs, where we state some problems for further research

New results on word-representable graphs

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 important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.], in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of $n$-vertex word-representable graphs is $2^{\frac{n^2}{3}+o(n^2)}$