5 research outputs found
Computing maximum cliques in -EPG graphs
EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection
graphs of paths on an orthogonal grid. The class -EPG is the subclass of
EPG graphs where the path on the grid associated to each vertex has at most
bends. Epstein et al. showed in 2013 that computing a maximum clique in
-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the
number of bends is at least , the class contains -interval graphs for
which computing a maximum clique is an NP-hard problem. The complexity status
of the Maximum Clique problem remains open for and -EPG graphs. In
this paper, we show that we can compute a maximum clique in polynomial time in
-EPG graphs given a representation of the graph.
Moreover, we show that a simple counting argument provides a
-approximation for the coloring problem on -EPG graphs without
knowing the representation of the graph. It generalizes a result of [Epstein et
al, 2013] on -EPG graphs (where the representation was needed)
Relationship of -Bend and Monotonic -Bend Edge Intersection Graphs of Paths on a Grid
If a graph can be represented by means of paths on a grid, such that each
vertex of corresponds to one path on the grid and two vertices of are
adjacent if and only if the corresponding paths share a grid edge, then this
graph is called EPG and the representation is called EPG representation. A
-bend EPG representation is an EPG representation in which each path has at
most bends. The class of all graphs that have a -bend EPG representation
is denoted by . is the class of all graphs that have a
monotonic (each path is ascending in both columns and rows) -bend EPG
representation.
It is known that holds for . We prove that
holds also for and for by investigating the -membership and -membership of complete
bipartite graphs. In particular we derive necessary conditions for this
membership that have to be fulfilled by , and , where and are
the number of vertices on the two partition classes of the bipartite graph. We
conjecture that holds also for .
Furthermore we show that holds for all
. This implies that restricting the shape of the paths can lead
to a significant increase of the number of bends needed in an EPG
representation. So far no bounds on the amount of that increase were known. We
prove that holds, providing the first result of this
kind
Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid
A graph is called an edge intersection graph of paths on a grid if there
is a grid and there is a set of paths on this grid, such that the vertices of
correspond to the paths and two vertices of are adjacent if and only if
the corresponding paths share a grid edge. Such a representation is called an
EPG representation of . is the class of graphs for which there
exists an EPG representation where every path has at most bends. The bend
number of a graph is the smallest natural number for which
belongs to . is the subclass of containing all graphs
for which there exists an EPG representation where every path has at most
bends and is monotonic, i.e. it is ascending in both columns and rows. The
monotonic bend number of a graph is the smallest natural number
for which belongs to . Edge intersection graphs of paths on a
grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of
research has been done on them since then.
In this paper we deal with the monotonic bend number of outerplanar graphs.
We show that holds for every outerplanar graph .
Moreover, we characterize in terms of forbidden subgraphs the maximal
outerplanar graphs and the cacti with (monotonic) bend number equal to ,
and . As a consequence we show that for any maximal outerplanar graph and
any cactus a (monotonic) EPG representation with the smallest possible number
of bends can be constructed in a time which is polynomial in the number of
vertices of the graph
Um estudo sobre grafos B2-EPG e B2-EPG-Helly
The word EPG is an acronym for Edge-Intersecting Paths on a Grid, that is, it exactly represents the class of edge-intersecion graphs of paths on a grid. In this writing, we started exploring the EPG graph subclass, well known as B2-EPG-Helly (more specifically its recognition complexity). However, we have also investigated graph representations that are not B1-EPG, but have not yet been associated with any Bk-EPG class, and also study other path properties in B2-EPG and B2-EPG with the Helly property. This research contains initial unpublished results about an exploration of the class B2- EPG, in addition to proposing interesting topics for future workA palavra EPG é um acrônimo para Edge-intersection Paths on a Grid, isto é, representa exatamente a classe de grafos de aresta-interseção de caminhos sobre uma grade. Neste trabalho de conclusão de curso iremos explorar principalmente uma subclasse de grafos EPG, conhecida como B2-EPG-Helly (mais especificamente a sua complexidade de reconhecimento). Contudo, também investigamos representações de grafos que não são B1-EPG, mas ainda não foram associadas a alguma classe Bk- EPG, além de estudar outras propriedades de caminhos em B2-EPG e B2-EPG com a propriedade Helly. Essa pesquisa contém resultados iniciais inéditos sobre a exploração da classe B2- EPG, além de propor tópicos interessantes para trabalhos futuro