6,624 research outputs found
Recognizing Cartesian graph bundles
AbstractGraph bundles generalize the notion of covering graphs and graph products. In this paper we extend some of the methods for recognizing Cartesian product graphs to graph bundles. Two main notions are used. The first one is the well-known equivalence relation δ★ defined on the edge-set of a graph. The second one is the concept of k-convex subgraphs. A subgraph H is k-convex in G, if for any two vertices x and y of distance d, d ⩽ k, each shortest path from x to y in G is contained entirely in H. The main result is an algorithm that finds a representation as a nontrivial Cartesian graph bundle for all graphs that are Cartesian graph bundles over a triangle-free simple base. The problem of recognizing graph bundles over a base containing triangles remains open
Cartesian product of hypergraphs: properties and algorithms
Cartesian products of graphs have been studied extensively since the 1960s.
They make it possible to decrease the algorithmic complexity of problems by
using the factorization of the product. Hypergraphs were introduced as a
generalization of graphs and the definition of Cartesian products extends
naturally to them. In this paper, we give new properties and algorithms
concerning coloring aspects of Cartesian products of hypergraphs. We also
extend a classical prime factorization algorithm initially designed for graphs
to connected conformal hypergraphs using 2-sections of hypergraphs
Square Property, Equitable Partitions, and Product-like Graphs
Equivalence relations on the edge set of a graph that satisfy restrictive
conditions on chordless squares play a crucial role in the theory of Cartesian
graph products and graph bundles. We show here that such relations in a natural
way induce equitable partitions on the vertex set of , which in turn give
rise to quotient graphs that can have a rich product structure even if
itself is prime.Comment: 20 pages, 6 figure
Fast Recognition of Partial Star Products and Quasi Cartesian Products
This paper is concerned with the fast computation of a relation on the
edge set of connected graphs that plays a decisive role in the recognition of
approximate Cartesian products, the weak reconstruction of Cartesian products,
and the recognition of Cartesian graph bundles with a triangle free basis.
A special case of is the relation , whose convex closure
yields the product relation that induces the prime factor
decomposition of connected graphs with respect to the Cartesian product. For
the construction of so-called Partial Star Products are of particular
interest. Several special data structures are used that allow to compute
Partial Star Products in constant time. These computations are tuned to the
recognition of approximate graph products, but also lead to a linear time
algorithm for the computation of for graphs with maximum bounded
degree.
Furthermore, we define \emph{quasi Cartesian products} as graphs with
non-trivial . We provide several examples, and show that quasi
Cartesian products can be recognized in linear time for graphs with bounded
maximum degree. Finally, we note that quasi products can be recognized in
sublinear time with a parallelized algorithm
The edge fault-diameter of Cartesian graph bundles
AbstractA Cartesian graph bundle is a generalization of a graph covering and a Cartesian graph product. Let G be a kG-edge connected graph and D̄c(G) be the largest diameter of subgraphs of G obtained by deleting c<kG edges. We prove that D̄a+b+1(G)≤D̄a(F)+D̄b(B)+1 if G is a graph bundle with fibre F over base B, a<kF, and b<kB. As an auxiliary result we prove that the edge-connectivity of graph bundle G is at least kF+kB
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