Intersection Graphs

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Intersection Graphs

• N.S.Narayanaswamy
• CSE Department
• IIT Madras

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Intersection Graphs

• Graph on n-vertices
• A set U, and n subsets of U assigned to vertices
  such that a pair of vertices is an edge if and
  only if the corresponding sets have a non-empty
  intersection
• Every graph is an intersection graph
• Define U to be set of edge labels
• Assign to each vertex the set of edge-labels of
  the edges incident on it.

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More structured sets?

• Intersection graphs of intervals on the real line
• Discs/rectangles in the plane
• Arcs of a circle
• Chords of a circle
• Lines in the plane
• Paths in a graph
• Sub-trees of a tree..

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Questions of interest

• Does the restricted nature of the set system
  impose a structural constraint on the
  corresponding intersection graph?
• Given a graph, is there a constructive way to
  identify the corresponding set-system, if one
  exists?
• What more can we obtain?
• Algorithms
• Other mathematical relationships

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Graph Theoretic Issues

• Is the intersection graph of a set of lines also
  the intersection graph of a set of axis parallel
  lines?
• Paths in a tree T and sub-trees of a another
  tree?
• Interval of real numbers and intervals of
  integers?
• Intervals of real numbers and intervals of unit
  length?
• Many more

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Sub-trees of a Tree

• T is a tree and V is a vertex set.
• For each v in V, let Tv be a sub-tree of T
• G is the intersection graph
• How large can an induced cycle be?
• Three
• Sub-trees of a tree- Induced cycle of length 4
  is impossible.
• Is this a sufficient condition?

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Chordal Graph

• Every cycle of length at least 4 has a chord
• Of a cycle is an edge between two non consecutive
  vertices.
• Each minimal vertex separator is a clique.
• A pair of non-adjacent vertices in a minimal
  vertex separator results in an induced cycle of 4
  vertices

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Simplicial vertices

• Neighbourhood of a vertex induces a clique.
• A chordal graph has two non-adjacent simplicial
  vertices
• Removing a simplicial vertex in a chordal graph
  results in a chordal graph
• Chordality is a hereditary property
• v1,v2,,vn such that vi is simplicial in G \
  v1,,vi-1 - perfect elimination ordering.

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Aside

• Characterization of chordal graphs
• Efficient test of chordality
• A modification of BFS
• Greedy coloring according to a PEO is the optimal
  coloring
• Recall Graph Coloring is NPC and it is widely
  believed that P and NP are different.
• Back to intersection graph of sub-trees of a tree

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Chordality is sufficient

• Given a chordal graph G, we construct a tree T
  and a set of sub-trees Tv, v ? G whose
  intersection graph is G.
• What is the tree if G is a clique-
• A single vertex and the for each vertex the
  associated tree is the tree itself. Clique.
• Why are we looking at the clique as a base case?
• The PEO can be seen as constructing G starting
  from a clique.

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Setting up the Induction

• Select a simplicial vertex v and let Kv be the
  maximal clique containing v
• Consider the set X of those vertices u such that
  u ? N(u) ? Kv
• X is non-empty as v is in X X is a clique.
• Let Y Kv\X
• Remove U from G and let T be the tree and Tu, u
  ? G\U be sub-trees that give G\U.

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Completing the Induction

• Let B a maximal clique in G\U containing Y.
• Case I If BY
• Rename the corresponding node in T as Kv
• In other words view this as adding X to B.
• Case II If B?Y
• Add a new node to T whose neighbour is B
• Label this new node as Kv

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The tree and the sub-trees

• The tree T is obtained from T
• Sub-trees-two cases again
• Case I If BY
• No change in sub-trees associated with vertices
  in G\U. not exactly rename B as Kv.
• sub-tree of vertices in U is the clique Kv

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The other case

• Case IIIf B?Y
• For each vertex in Y, add Kv to its sub-tree.
• For all other vertices in G\U, the sub-tree is
  unchanged
• For all vertices in U, the sub-tree is Kv
• In either case the edges of G\U are captured by
  sub-tree intersection and the edges incident on v
  are captured by intersections with Kv. No
  spurious intersections.

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Consequences

• View chordal graphs as intersection graph of
  sub-trees of a tree.
• Helps in algorithms. For example, FPT algorithms
  for minimum dominating sets in chordal graphs
  SWAT 2006
• Intersection graph of sub-trees of a path(special
  kind of a tree) is an interval graph
• A characterization

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Connectivity

• Minimal vertex separator
• 1,6,10 disconnects 4
• Connectivity size of minimum vertex separator
• Connectedness K-connected graph has
  connectivity exactly k
• Eg 3-connected graph

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Contractible edge

• Contraction does not decrease the connectivity

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Research Questions

• Characterization
• Bounds on number of Contractible edges
• Distribution of Contractible edges
• Many papers- Tutte, Saito, Dean etc.
• Results exist up to 8-connected graphs
• Not many results on special graph classes - How
  about chordal graphs?

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Tree decomposition PEO
PEO 10 11 9 8 6 5 7 4 3 2 1
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Bi connectivity of contractible edges

• In a chordal graph, contractible edges form a
  bi-connected graph.
• Connectivity is 2

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A characterization

• eu,v is contractible iff one of the following
  is true
• e is in a unique maximal clique in G
• For x,y in V(T), such that x,y ?E(T),
  u,v?l(x)?l(y) and size of l(x)?l(y) is greater
  than k

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Example
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Example