12 research outputs found
Planar graphs as L-intersection or L-contact graphs
The L-intersection graphs are the graphs that have a representation as
intersection graphs of axis parallel shapes in the plane. A subfamily of these
graphs are {L, |, --}-contact graphs which are the contact graphs of axis
parallel L, |, and -- shapes in the plane. We prove here two results that were
conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are
L-intersection graphs, and that triangle-free planar graphs are {L, |,
--}-contact graphs. These results are obtained by a new and simple
decomposition technique for 4-connected triangulations. Our results also
provide a much simpler proof of the known fact that planar graphs are segment
intersection graphs
On some special classes of contact -VPG graphs
A graph is a -VPG graph if one can associate a path on a rectangular
grid with each vertex such that two vertices are adjacent if and only if the
corresponding paths intersect at at least one grid-point. A graph is a
contact -VPG graph if it is a -VPG graph admitting a representation
with no two paths crossing and no two paths sharing an edge of the grid. In
this paper, we present a minimal forbidden induced subgraph characterisation of
contact -VPG graphs within four special graph classes: chordal graphs,
tree-cographs, -tidy graphs and -free graphs. Moreover, we present a
polynomial-time algorithm for recognising chordal contact -VPG graphs.Comment: 34 pages, 15 figure
Characterising circular-arc contact -VPG graphs
A contact -VPG graph is a graph for which there exists a collection of
nontrivial pairwise interiorly disjoint horizontal and vertical segments in
one-to-one correspondence with its vertex set such that two vertices are
adjacent if and only if the corresponding segments touch. It was shown by Deniz
et al. that Recognition is -complete for contact -VPG graphs.
In this paper we present a minimal forbidden induced subgraph characterisation
of contact -VPG graphs within the class of circular-arc graphs and provide
a polynomial-time algorithm for recognising these graphs
Characterising Chordal Contact: Bo-VPG Graphs
A graph G is a Bo- VPG graph if it is the vertex intersection graph of horizontal and vertical paths on a grid. A graph G is a contact Bo- VPG graph if the vertices can be represented by interiorly disjoint horizontal or vertical paths on a grid and two vertices are adjacent if and only if the corresponding paths touch. In this paper, we present a minimal forbidden induced subgraph characterisation of contact Bo-VPG graphs within the class of chordal graphs and provide a polynomial-time algorithm for recognising these graphs
On contact graphs of paths on a grid
In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0- CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG
The maximum number of edges in a graph of bounded dimension, with applications to ring theory
AbstractWith a finite graph G = (V, E), we associate a partially ordered set P = (X, P) with X = V ∪ E and x < e in P if and only if x is an endpoint of e in G. This poset is called the incidence poset of G. In this paper, we consider the function M(p, d) defined for p, d ⩾ 2 as the maximum number of edges a graph G can have when it has p vertices and the dimension of its incidence poset is at most d. It is easy to see that M(p, 2) = p − 1 as only the subgraphs of paths have incidence posets with dimension at most 2. Also, a well known theorem of Schnyder asserts that a graph is planar if and only if its incidence poset has dimension at most 3. So M(p, 3) = 3 p − 6 for all p ⩾ 3. In this paper, we use the product ramsey theorem, Turán's theorem and the Erdős/Stone theorem to show that limp→∞ M(p, 4)/p2 = 38. We then derive some ring theoretic consequences of this in terms of minimal first syzygies and Betti numbers for monomial ideals
On the helly property of some intersection graphs
An EPG graph G is an edge-intersection graph of paths on a grid. In this
doctoral thesis we will mainly explore the EPG graphs, in particular B1-EPG graphs.
However, other classes of intersection graphs will be studied such as VPG, EPT and
VPT graph classes, in addition to the parameters Helly number and strong Helly
number to EPG and VPG graphs. We will present the proof of NP-completeness
to Helly-B1-EPG graph recognition problem. We investigate the parameters Helly
number and the strong Helly number in both graph classes, EPG and VPG in order
to determine lower bounds and upper bounds for this parameters. We completely
solve the problem of determining the Helly and strong Helly numbers, for Bk-EPG,
and Bk-VPG graphs, for each value k.
Next, we present the result that every Chordal B1-EPG graph is simultaneously
in the VPT and EPT graph classes. In particular, we describe structures that occur
in B1-EPG graphs that do not support a Helly-B1-EPG representation and thus we
define some sets of subgraphs that delimit Helly subfamilies. In addition, features
of some non-trivial graph families that are properly contained in Helly-B1 EPG are
also presented.EPG é um grafo de aresta-interseção de caminhos sobre uma grade.
Nesta tese de doutorado exploraremos principalmente os grafos EPG, em particular
os grafos B1-EPG. Entretanto, outras classes de grafos de interseção serão estu dadas, como as classes de grafos VPG, EPT e VPT, além dos parâmetros número
de Helly e número de Helly forte nos grafos EPG e VPG. Apresentaremos uma
prova de NP-completude para o problema de reconhecimento de grafos B1-EPG Helly. Investigamos os parâmetros número de Helly e o número de Helly forte nessas
duas classes de grafos, EPG e VPG, a fim de determinar limites inferiores e superi ores para esses parâmetros. Resolvemos completamente o problema de determinar o
número de Helly e o número de Helly forte para os grafos Bk-EPG e Bk-VPG, para
cada valor k.
Em seguida, apresentamos o resultado de que todo grafo B1-EPG Chordal está
simultaneamente nas classes de grafos VPT e EPT. Em particular, descrevemos
estruturas que ocorrem em grafos B1-EPG que não suportam uma representação
B1-EPG-Helly e assim definimos alguns conjuntos de subgrafos que delimitam sub famílias Helly. Além disso, também são apresentadas características de algumas
famílias de grafos não triviais que estão propriamente contidas em B1-EPG-Hell