Circle Graph and Circular Arc Graph Recognition

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Circle Graph and Circular Arc Graph Recognition
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Outlines

• Circle Graph Recognition
• Circular-Arc Graph Recognition

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Chords of a circle

• A chord v is associated two endpoints x,y

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Circle Graph

• A circle graph is a graph whose vertices can be
  associated with chords of a circle such that two
  vertices are adjacent if and only if the
  corresponding chords in the circle intersect.

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Unique Chord Models

• Consider the two equivalent Chord Models of the
  following circle graph.
• A circle graph is uniquely represented if all of
  its Chord Models are equivalent.

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Circle Graph Recognition
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Overview

• Let G(V,E) be the input circle graph.
• Decompose a given graph G into the Prime
  Subgraphs through join decomposition
• Each prime component has a unique chord model
• Create a a unique chord model for a Prime
  Subgraph
• Find a member W of a family F whose members have
  unique chord models first.
• Determine the chords for the vertices in V-W.

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Join Decomposition
V3
V0
V1
V2
m1
m2
All Edges
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Prime Subgraphs

• A subgraph G(V, E) is a prime subgraph iff it
  is join-inseperatable.
• A prime subgraph G, V 5, contains a subgraph
  (a member of the set F) which has a simple and
  unique chord presentation.
• The family of the subgraph which has a simple and
  unique chord presentation is named as the set F.

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Members of the Set F
(b) A tepee
(c) A figure-8
(d) Primitive cycles of length k ? 5
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The Members of the Set F
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The unique placement of chords

• Assume we have found a W which is a member of f.
  We construct the unique chord model for W first.
• Then, if we arrange the remaining vertices
  carefully, there is a unique placement for each
  of them iteratively. (C. Gabor, W. L. Hsu and K.
  Supowit, Recognizing circle graphs in polynomial
  time," J. Assoc. Comput. Machin., 435-473, (1989)
  )

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Circular-Arc Graph Recognition

• Check the flow chart
• in the handout

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Circular-Arc Graph

• A circular-arc graph is the intersection graph of
  a set of arcs on the circle. It has one vertex
  for each arc in the set, and an edge between
  every pair of vertices corresponding to arcs that
  intersect.

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An Arc Model

• Denote arc i by head(i), tail(i) scanned in the
  clockwise order.

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Two Different Arc Modelsunnecessary variations
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Vertex Adjacency in circular arc graphs

• Two vertices v1 and v2 in G are said to be
• Independent if v1 is not adjacent to v2.
• strictly adjacent if v1 is adjacent to v2 but
  neither N(v1) nor N(v2) is contained in the
  other.
• strongly adjacent if v1 and v2 are strictly
  adjacent and every w in V(G)\N(v1) satisfies that
  N(w) ? N(v2) and every w in V(G)\N(v2)
  satisfies that N(w) ? N(v1) .
• Similar if N(v1)\v1N(v2)\v2
• To define normalized models, we need to assume
  there are no similar vertices

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Overlapping relationships of a pair of arcs
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A normalized Arc Model

• An arc Model R for a Circular-Arc Graph G is
  normalized if every pair of arcs u1 and u2 and
  the corresponding pair of nodes v1and v2 satisfy
• (1) u1 is independent of u2 ltgt v1 is not
  adjacent to v2
• (2) u1 is contained in u2 ltgt N(v1) ? N(v2)
• (3) u1 strictly overlaps u2 ltgt v1 is strictly
  but not strongly adjacent to v2
• (4) u1 and u2 cover the circle ltgt v1 is strictly
  but not strongly adjacent to v2

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A Unique Normalized Model

• A first step towards unique arc model.
• To avoid unnecessary model variation, define a
  Normalized Model for each circular-arc graph
• An arc model C is equivalent to C if C can be
  obtained from C by rotation (shift) and
  reflection (clockwise ? counter-clockwise).
• A circular-arc graph is said to have a unique
  normalized model if all normalized models are
  equivalent.

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How to normalize an existing arc model?
Possible Violations

• Type I
• u1 strictly overlaps u2 but v1 and v2 are
  strongly adjacent. (Algorithm I)
• Type II
• u1 strictly overlaps u2 but v1 is contained in
  v2. (Algorithm II)

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Two types of Violations

• If R is not a normalized model for G, there are
  two types of violations.
• Type I
• u1 strictly overlaps u2 but v1 and v2 are
  strongly adjacent. (eliminated by Algorithm I)
• Type II
• u1 strictly overlaps u2 but v1 is contained in
  v2. (eliminated by Algorithm II)

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Endpoint Blocks of an Arc Model

• A head (or tail) block is a set of maximal
  contiguous subsequence of heads (tails)

h(1)
t(1)
1 2 3 4
h(2)
t(2)
h(3)
h(4)
t(3)
t(4)
head block H
tail block T
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Algorithm I (eliminating Type I violation) 1

• For each head h(i), find the first tail block
  T(h(i)) by a counterclockwise traversal from h(i)

t(i)
h(i)
T(h(i))
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Algorithm I (eliminating Type I violation) 2

• T(h(i)) T1?T2. T1 is the tails of those
  corresponding arcs not overlapping arc i.
• T2 T(h(i))\T1

T2
T1
t(i)
h(i)
T(h(i))
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Algorithm I (eliminating Type I violation) 3

• If T2 is not empty, insert h(i) between T2 and
  T1.
• Repeat the procedure for t(i).

T1
T2
t(i)
h(i)
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Algorithm II (eliminating Type II violation)

• For each head block, sort the order of the heads
  by its corresponding reverse order of tails.

1 2 3 4
h(4) lt h(2) gt h(1) gt h(3)
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The Associated Circle Graph Gc

• Associate with each graph G the graph Gc that has
  the same vertex set as G such that two vertices
  in Gc are adjacent iff they are strictly but not
  strongly adjacent in G.

Gc
G
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Utilizing theAssociated Circle Graph Gc

• G is a circular-arc graph iff all induced
  subgraphs of G are circular arc graph.
• Use the Associated Circle Graph Gc to construct a
  normalized arc model of G.
• However, the same arc model may still associate
  with multiple chords model.
• How to deal with such a phenomenon? Graph
  decomposition

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The non-Unique Chord Representation (Type I)
Consider the connected components of Gc
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The non-Unique Chord Representation (Type II)
Consider the module-free subgraph of Gc
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Modular Decomposition

• To obtain components which do not contain type I
  or type II structures, can use Modular
  Decomposition to decompose Gc

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Conformal Model

• Does the chord model necessarily represent a
  N-model?
• The chord model represents a N-model if it is a
  Conformal Model.
• Three chords d1, d2 and d3 (d1 d2 d3 ) are
  said to be parallel if

series
parallel
d2
d1
d3
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Conformal Model

• Three nodes vi,vj and vk of G are parallel
  (vivjvk) if one of the following 8 conditions
  is satisfied
• (1) vk contains vj vj contains vi.
• (2) vk contains vj vj and vi cover the circle.
• (3) vk is contained in vj vi contains vj.
• (4) vk is contained in vj vi is independent of
  vj.
• (5) vk is independent of vj vj contains vi.
• (6) vk is independent of vj vj and vi cover the
  circle.
• (7) vk and vj cover the circle vj is independent
  of vi
• (8) vk and vj cover the circle vi contains vj.

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Conformal Model

• A chord model is a Conformal Model iff v1v2v3
  when d1d2d3.
• Let G be a circular-arc graph with a connected
  Gc (the complement of Gc). Then a model for Gc
  is conformal iff it is a chord model associated
  with an N-model for G.

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The Transformation from chords back to Arcs (easy
part)

• Let G be a circular-arc graph with a connected
  Gc (the complement of Gc). Let D be an
  associated chord model of Gc. Then D determines a
  unique N-model R of G.
• For a fixed arc i, a new arc j
• Is on the opposite side of i if i is independent
  of j
• or cover the circle with j.
• A new arc j on the same side of i if i contains j
  or is contained in j.

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The Transformation from chords back to Arcs
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Gc
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The complement of Gc
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Summary

• Recognition Algorithm
• Take a proposed circular-arc graph G and
  transform it into a circle graph Gc
• Decompose Gc into s-inseparable components.
• If G is indeed a circular-arc graph, we can find
  a conformal model of Gc and transform it back to
  an N-model of G.
• Otherwise, there is a contradiction, and G is not
  a circular arc graph.