Visualization of graphs using their product structure

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Visualization of graphs using their product
structure

• Bla Zmazek and Janez erovnik
• FME University of Maribor and
• IMFM

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Introduction

• Knowledge of the structure of a graph
• 1. Faster algorithms for solving problems
  (Examples chromatic number, )
• 2. Visualization of the graph
• Graph drawing (conferences, competitions, )
• CLAIM if known, structural properties can help
• EXAMPLE graph products and bundles

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References

• Graph drawing
• vast literature, Batagelj and Mrvar won 2005
  competition at Graph Drawing
• .
• Graph products
• Imrich, Klavar Product Graphs, structure and
  recognition, Wiley 2000.
• Graph bundles
• Pisanski, Shawe-Taylor, Vrabec JCT (B) 1983
• .
• Less studied, some recent results by Pisanski,
  Imrich, Zmazek, erovnik,

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Definitions

• Graph
• Graph products
• Cartesian
• Direct
• Strong
• Lexicographic

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Cartesian product

• is up to isomorphism determined by its factors
  (the order being unimportant)
• Examples
• squares,
• hypercubes,
• prisms (Cartesian
• products of n-gons by an edge),
• the square lattice as the product of two infinite
  paths.

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Cartesian product

• All finite connected graphs have unique prime
  factorizations with respect to the Cartesian
  product
• Product relation (Sabidussi)
• Djokovic local product relation (lpr)
• convex hull of the lpr
• ? recognition in polynomial time

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Applications ofCartesian product

• Hamming graphs
• used in coding theory
• and median graphs are isometric subgraphs of
  Cartesian products
• their recognition is fast
• design of interconnection networks
• Hypercubes
• Reconstruction of a Cartesian product
• the problem is polynomial

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Other products

• Direct product
  polynomial algorithm for recognition (Imrich)
• The set of edges of the strong product is the
  union of edges of Cartesian product and edges of
  direct product.
  polynomial algorithm for recognition (Feigenbaum
  at al.)
• Feigenbaum and Schaffer the problem of
  recognizing composite graphs with respect to the
  lexicographic product is equivalent to testing
  graph isomorphism

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(Cartesian) graph bundle
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Graph bundles

• generalize the notion of
• covering graphs (NP-hard recognition)
• graph products
• more than one presentation of graph as a graph
  bundle may exist
• Cartesian and strong bundles with totally
  disconnected fibre
• covering graphs

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Cartesian graph bundle

• All presentations can be found in polynomial time
  for a Cartesian graph bundle over base graph with
  at least one minimal presentation with fibre
  different from K2 Zmazek, erovnik
• There is a polynomial algorithm for recognizing
  Cartesian graph bundles over
• triangle-free base,
• triangles disjoint in base and
• K4 \ e-free base

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Examples
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Examples
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