7 research outputs found
On the Representability of Line Graphs
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) is in E for each
x not equal to y. The motivation to study representable graphs came from
algebra, but this subject is interesting from graph theoretical, computer
science, and combinatorics on words points of view. In this paper, we prove
that for n greater than 3, the line graph of an n-wheel is non-representable.
This not only provides a new construction of non-representable graphs, but also
answers an open question on representability of the line graph of the 5-wheel,
the minimal non-representable graph. Moreover, we show that for n greater than
4, the line graph of the complete graph is also non-representable. We then use
these facts to prove that given a graph G which is not a cycle, a path or a
claw graph, the graph obtained by taking the line graph of G k-times is
guaranteed to be non-representable for k greater than 3.Comment: 10 pages, 5 figure
New results on word-representable graphs
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
for each . 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 -vertex word-representable graphs is
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
Word-representability of face subdivisions of triangular grid 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) ∈ E. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph K4. Inspired by a recent elegant result of Akrobotu et al., who classified wordrepresentable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpi´nski gasket graph is wordrepresentable
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
Graph parameters and the speed of hereditary properties
In this thesis we study the speed of hereditary properties of graphs and how this defines some of the structure of the properties. We start by characterizing several graph parameters by means of minimal hereditary classes. We then give a global characterization of properties of low speed, before looking at properties with higher speeds starting at the Bell number. We then introduce a new parameter, clique-width, and show that there are an infinite amount of minimal hereditary properties with unbounded clique-width. We then look at the factorial layer in more detail and focus on P7-free bipartite graphs. Finally we discuss word-representable graphs