1,353 research outputs found
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
Crossing Patterns in Nonplanar Road Networks
We define the crossing graph of a given embedded graph (such as a road
network) to be a graph with a vertex for each edge of the embedding, with two
crossing graph vertices adjacent when the corresponding two edges of the
embedding cross each other. In this paper, we study the sparsity properties of
crossing graphs of real-world road networks. We show that, in large road
networks (the Urban Road Network Dataset), the crossing graphs have connected
components that are primarily trees, and that the remaining non-tree components
are typically sparse (technically, that they have bounded degeneracy). We prove
theoretically that when an embedded graph has a sparse crossing graph, it has
other desirable properties that lead to fast algorithms for shortest paths and
other algorithms important in geographic information systems. Notably, these
graphs have polynomial expansion, meaning that they and all their subgraphs
have small separators.Comment: 9 pages, 4 figures. To appear at the 25th ACM SIGSPATIAL
International Conference on Advances in Geographic Information Systems(ACM
SIGSPATIAL 2017
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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)
Constructing dense graphs with sublinear Hadwiger number
Mader asked to explicitly construct dense graphs for which the size of the
largest clique minor is sublinear in the number of vertices. Such graphs exist
as a random graph almost surely has this property. This question and variants
were popularized by Thomason over several articles. We answer these questions
by showing how to explicitly construct such graphs using blow-ups of small
graphs with this property. This leads to the study of a fractional variant of
the clique minor number, which may be of independent interest.Comment: 10 page
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
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