3,207 research outputs found
On edge intersection graphs of paths with 2 bends
An EPG-representation of a graph G is a collection of paths in a grid, each corresponding to a single vertex of G, so that two vertices are adjacent if and only if their corresponding paths share infinitely many points. In this paper we focus on graphs admitting EPG-representations by paths with at most 2 bends. We show hardness of the recognition problem for this class of graphs, along with some subclasses. We also initiate the study of graphs representable by unaligned polylines, and by polylines, whose every segment is parallel to one of prescribed slopes. We show hardness of recognition and explore the trade-off between the number of bends and the number of slopes. © Springer-Verlag GmbH Germany 2016
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
On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can
be represented as the edge intersection graph of paths on a grid (EPG graph),
i.e., one can associate with each vertex of the graph a nontrivial path on a
rectangular grid such that two vertices are adjacent if and only if the
corresponding paths share at least one edge of the grid. For a nonnegative
integer , -EPG graphs are defined as EPG graphs admitting a model in
which each path has at most bends. Circular-arc graphs are intersection
graphs of open arcs of a circle. It is easy to see that every circular-arc
graph is a -EPG graph, by embedding the circle into a rectangle of the
grid. In this paper, we prove that every circular-arc graph is -EPG, and
that there exist circular-arc graphs which are not -EPG. If we restrict
ourselves to rectangular representations (i.e., the union of the paths used in
the model is contained in a rectangle of the grid), we obtain EPR (edge
intersection of path in a rectangle) representations. We may define -EPR
graphs, , the same way as -EPG graphs. Circular-arc graphs are
clearly -EPR graphs and we will show that there exist circular-arc graphs
that are not -EPR graphs. We also show that normal circular-arc graphs are
-EPR graphs and that there exist normal circular-arc graphs that are not
-EPR graphs. Finally, we characterize -EPR graphs by a family of
minimal forbidden induced subgraphs, and show that they form a subclass of
normal Helly circular-arc graphs
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)
1-String CZ-Representation of Planar Graphs
In this paper, we prove that every planar 4-connected graph has a
CZ-representation---a string representation using paths in a rectangular grid
that contain at most one vertical segment. Furthermore, two paths representing
vertices intersect precisely once whenever there is an edge between
and . The required size of the grid is
Proper circular arc graphs as intersection graphs of paths on a grid
In this paper we present a characterisation, by an infinite family of minimal
forbidden induced subgraphs, of proper circular arc graphs which are
intersection graphs of paths on a grid, where each path has at most one bend
(turn)
- …