Edgeclique Graphs of Chordal Graphs

1
Edge-clique Graphs of Chordal Graphs

• Kit Barton
• CMPUT 672 Project
• March 30, 2004

2
Overview

• Edge-clique covers and Edge-clique graph
• Previous Work
• Edge-clique graphs
• Algorithm
• Edge clique covers
• Conclusions

3
Edge Clique Cover (ECC)

• Given a graph G, find a set of cliques such that
  every edge of G is included in at least one clique

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Edge Clique Graph (ECG)

• Provides a way to represent the ECC of a graph
• Nodes in Ke(G) correspond to edges in G
• Two nodes are connected if and only if their
  corresponding edges in G belong to the same clique

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Example
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Previous Work

• Kou, Stockmeyer and Wong (1978)
• ECC problem and the keyword conflict problem
• Albertson and Collins (1984)
• Formulated the ECG problem
• if G is chordal Ke(G) is also chordal

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Previous Work

• Raychudhuri (1988)
• ECC problem and the intersection number of a
  graph
• Quadratic algorithm to find the ECC of a chordal
  graph
• Other work that focuses on specific types of
  chordal graphs and their ECG

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Edge Clique Graphs

• Is every chordal graph is the Edge Clique graph
  of a chordal graph?
• No
• proof is based on Alberson and Collins
• observation

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Perfect Elimination Orderings of ECGs
ve3 ve2 ve5 ve4 ve1
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ve3 ve2 ve7 ve6 ve4 ve1 ve5
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Critical Vertices

• A critical vertex is a vertex contained in more
  than one maximal clique of Ke(G)
• A critical vertex in Ke(G) corresponds to a
  critical edge in G
• Given a perfect elimination ordering O of Ke(G),
  for every critical vertex vc contained in cliques
  c1...ck, no more than one of c1...ck will be
  located to the right of vc in O

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Algorithm Overview

• Use a perfect elimination ordering of vertices in
  G
• MaxDegreeLexBFS
• For each vertex v in G,
• find all edges leaving v and their corresponding
  vertices in Ke(G)
• connect all pairs of vertices in Ke(G)
• remember neighbours of v and the edges that
  connect them
• if v has been remembered, look up edges that are
  associated with it

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Data Structures

• Watch array
• used to remember vertices in perfect elimination
  ordering
• one-dimensional boolean array indexed by v
• ConnectList
• for a remembered vertex, list the neighbours of
  interest and corresponding ECG vertices
• two dimensional array indexed by v, containing
  ECG vertices

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Example
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Vertices
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Initial Conditions
ve1
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ConnectList
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First Iteration Step 1
ConnectList
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First Iteration Step 2
ConnectList
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v2
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ve2
ve3
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First Iteration Step 2
ve1
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ve5
ve3
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ConnectList
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v2
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First Iteration Step 3
v1
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F
F
F
F
Watch
ConnectList
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Second Iteration Step 1
ConnectList
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ve1
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Second Iteration Step 2
ConnectList
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Second Iteration Step 2
ve1
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ConnectList
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Second Iteration Step 3
ConnectList
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Third Iteration Step 1
ConnectList
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ve5
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Third Iteration Step 2
ve1
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ConnectList
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Algorithm Discussion

• All neighbours to the right of the current vertex
  form a clique
• Edges leaving current vertex must be connected
• Future edges must also be considered to form the
  proper cliques
• Watch and ConnectList are used to remember ECG
  vertices for future connections

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Algorithm Complexity

• Dominated by
• Edge comparisons
• Reduced by MaxDegreeLexBFS
• Still O(m2) in some cases
• Complete graph on k gt 3 vertices
• Vertex connections
• Required because of Edge-clique graph structure

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Edge Clique Covers

• Constructing an ECC of a chordal graph is hard
  because of critical edges

Vertices
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No efficient method for identifying and placing
critical edges could be found
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Conclusions

• Algorithm to construct the ECG of a chordal graph
• Performs O(m2) edge comparisons in worst case
• Performs O(m2) vertex connections in all cases
• Create ECG in one step
• Identified what makes find an ECC of a chordal
  graph difficult

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Future Work

• Identification of critical edges for ECC
  construction
• Necessary and sufficient conditions for a graph G
  to be an ECG(H)
• If G is in class X then ECG(G) is also in class X
  for classes of graphs other than chordal graphs

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Example 2
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Second Iteration
ve1
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ConnectList
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ve5
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Old Example
Vertices
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Old Example
ConnectList
v6
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ve8
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Old Example
ConnectList
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Old Example
ConnectList
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T
ve9
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Old Example
ve1
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ConnectList
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T
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Old Example
ConnectList
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Miscellaneous
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Algorithm Parts
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Algorithm Parts
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Old Example
for j 1 to d(vc) do
ej edge leaving vc
vj target of ej in G
vej V(ej) in Ke(G)
if Watchvc true
foreach vek in ConnectListvcvj
Connect vej to vek in Ke(G)
ConnectList
endfor
endif
v6
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v5
ve8
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Old Example