10 research outputs found
On graphs with representation number 3
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
is an edge in . A graph is word-representable if and only if it is
-word-representable for some , that is, if there exists a word containing
copies of each letter that represents the graph. Also, being
-word-representable implies being -word-representable. The minimum
such that a word-representable graph is -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
, is that the Petersen graph and triangular prism belong to this
class. In this paper, we show that any prism belongs to , and
that two particular operations of extending graphs preserve the property of
being in . Further, we show that is not included
in a class of -colorable graphs for a constant . 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
-word-representable, and thus each ladder graph is a circle graph. Finally,
we discuss -word-representing comparability graphs via consideration of
crown graphs, where we state some problems for further research
Semi-Transitive Orientations and Word-Representable Graphs
A graph is a \emph{word-representable graph} if there exists a word
over the alphabet such that letters and alternate in if and
only if for each .
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 is the minimum such that is a
representable by a word, where each letter occurs times; such a exists
for any word-representable graph. We show that the representation number of a
word-representable graph on vertices is at most , while there exist
graphs for which it is .Comment: arXiv admin note: text overlap with arXiv:0810.031
On word-representability of polyomino triangulations
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
is an edge in . 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 -cycle in a polyomino by the
complete graph is word-representable. We employ semi-transitive
orientations to obtain our results
Minimum length word-representants of graph products
A graph is said to be word-representable if a word can be
formed using the letters of the alphabet such that for every pair of
vertices and , if and only if and alternate in .
Gaetz and Ji have recently introduced the notion of minimum length
word-representants for word-representable graphs. They have also determined the
minimum possible length of the word-representants for certain classes of
graphs, such as trees and cycles. It is know that Cartesian and Rooted products
preserve word-representability. Moreover, Broere constructed a uniform word
representing the Cartesian product of and using occurrence based
functions.
In this paper, we study the minimum length of word-representants for
Cartesian and Rooted products using morphism and occurrence based function,
respectively. Also, we solve an open problem posed by Broere in his master
thesis. This problem asks to construct a word for the Cartesian product of two
arbitrary word-representable graphs.Comment: 14 pages, submitted to DA
Графы, представимые в виде слов : обзор результатов
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
Representing Graphs via Pattern Avoiding Words
The notion of a word-representable graph has been studied in a series of
papers in the literature. A graph is word-representable if there
exists a word over the alphabet such that letters and alternate
in if and only if is an edge in . If , this is
equivalent to saying that is word-representable if for all , if and only if the subword of
consisting of all occurrences of or in has no consecutive
occurrence of the pattern 11.
In this paper, we introduce the study of -representable graphs for any
word . A graph is -representable if and only if there
is a labeled version of , , and a word such that for all , if and
only if has no consecutive occurrence of the pattern . Thus,
word-representable graphs are just -representable graphs. We show that for
any , every finite graph is -representable. This contrasts
with the fact that not all graphs are 11-representable graphs.
The main focus of the paper is the study of -representable graphs. In
particular, we classify the -representable trees. We show that any
-representable graph is a comparability graph and the class of
-representable graphs include the classes of co-interval graphs and
permutation graphs. We also state a number of facts on -representation of
induced subgraphs of a grid graph
A comprehensive introduction to the theory of word-representable graphs
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 offers a comprehensive introduction to the theory of word-representable graphs including the most recent developments in the area
On the representation number of a crown graph
A graph G = (V,E) is word-representable if there exists a word ω over the alphabet V such that letters x and y alternate in ω if and only if xy is an edge in E . It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word ω representing G such that each letter occurs exactly k times in ω. The minimum such k is called G’s representation number. A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥ 5,Hn,n ’s representation number is [n / 2]. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number
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