Even Pairs and Perfect Graphs

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Even Pairs and Perfect Graphs

• Ravi Prakash
• under the guidance of
• Prof. S.Biswas
• Department of Computer Science and Engg.
• IIT Bombay
• April 11,2006

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Introduction

• Two non-adjacent vertices in a graph form an even
  pair if every chordless path between them has an
  even number of edges.
• Contraction For two vertices x ,y in a graph G,
  we denote by Gxy the graph obtained by deleting
  x and y and adding a new vertex xy adjacent to
  precisely those vertices of G- x-y which are
  adjacent to atlest one of x, y in G .We say that
  Gxy is obtained by contracting on the pair x,
  y.
• Silent fact about even pair is that contracting
  an even pair in a graph G yields a graph that has
  the same clique number and chromatic number as G.
• The contraction of two vertices that form an even
  pair in an perfect graph produces a new perfect
  graph.

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Notations

• ?(G) Chromatic number of G.
• ?(G) Size of largest clique in G.
• a(G) Size of largest stable set in G.
• Some Relationship
• ?(Gxy) ?(G)
• ?(G) ?(Gxy)
• Note The problem of computing X(G) and
  W(G) are in general NP-complete.
• Even-contractile graph If there is a
  sequence G0G,G1,....,Gj such that Gj is a
  clique, and, for i j-1,Gi1 is obtained from Gi
  via the contraction of an even pair of Gi.
• Perfectly contractile graph A graph is
  perfectly contractile if each of its induced
  subgraphs is even contractile.

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A sequence of even pair contraction
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• Introduction
• Definition Even Pair , Perfect Graph
• Notation Used
• Even Pair Lemma
• Perfectly Contractile Graph
• Hereditary class
• Hereditary class and
  Perfectly Contractile graph
• Weakly Triangulated
  Graphs
• Meyniel Graphs and
  quasi-Meyniel Graphs
• Perfectly Orderable
  Graphs
• Graphs That might be perfectly contractile
• Quasi-parity graphs
  (QP)
• Bull
  Free Graph
• Graphs
  with no P5 or K5
• Strictly Quasi-parity
  graphs
• Slim
  graph Slender graph Skeletal Graph

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Relation Even Pair and Perfect graphs

• Let x, y be an even pair in a perfect graph
  G. Then Gxy is perfect.
• Even pair lemma
• Implies that any graph G such that every induced
  subgraph of G (or its complement) has an even
  pair must be a perfect graph.

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Perfect graph with node4
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Graph with no even pair
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PERFECTLY CONTRACTILE GRAPH

• A class of graphs contains only perfectly
  contractile graphs means that every graph in the
  class is perfect.
• Hereditary class means every induced subgraph of
  a graph in the class is also in the class.
• If A is a hereditary class of graphs and every
  graph in A is either a clique or contains an even
  pair whose contraction yields a graph in A then
  every graph in A is perfectly contractile.

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To show hereditary class C of graphs contains
only perfectly contractile graphs

• Its not sufficient to prove that every
  non-complete graph in C has an even pair.
• Must show that every graph in C is the beginning
  of a sequence of even-pair contractions that
  leads to a clique.
• Important to note that the intermediate graphs of
  the sequence need not be in the class C.

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• Two situation can occur
• Given a graph G of C, find an even pair x,y of
  G such that Gxy is in C. Using the result in
  last slide we can show that G is perfectly
  contractile.
• C could contain a graph G such that all possible
  even-pair contractions from G produce graphs
  that are not in C.

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Weakly Triangulated graphs

• A graph is weakly triangulated if it contains no
  holes(Ck,k5) and no anti- holes (complement of
  Ck,k5).
• Every weakly triangulated graph is perfect.
• Every weakly triangulated graph contains an even
  pair.
• Its not true that contracting an even pair in a
  weakly triangulated graph always yields a new
  weakly triangulated graph.

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• Contracting a 2-pair in a weakly triangulated
  graph yields a weakly triangulated graph.
• Every weakly triangulated graph which is not a
  clique contains a 2-pair.

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Meyniel Graph

• A Meyniel Graph is any graph in which every odd
  cycle of length atleast five has two chords.
• These graphs are perfect i,e has either a clique
  or contains an even pair.
• quasi-Meyniel Graph
• Every Meyniel is quasi-Meyniel.

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• If G is quasi-Meyniel but not Meyniel,then G has
  atmost two tips and if x is a tip of G,then G-x
  is Meyniel.
• Every non-complete quasi-Meyniel graph contains
  an even pair whose contraction yields a
  quasi-Meyniel graph.

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Perfectly orderable graph

• An order is perfect if and only if G contains no
  path P4 on four vertices w,x,y,z with edges wx,
  xy, yz such that w
• A graph is perfectly orderable if it admits a
  perfect order.
• Every non-complete perfectly orderable graph has
  an even pair.

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• Perfectly orderable graphs are perfectly
  contractile.

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Graphs that might be perfectly contractile

• Strongly perfect graphs
• That is if every induced subgraph H has an
  independent set meeting all maximal cliques of H.
• A graph is perfect if and only if each of its
  induced subgraphs contains a stable set meeting
  all maximum cliques.
• Strongly perfect graphs are perfect.

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Alternately orientable graphs

• A graph is alternately orientable iff it
  admits an orientation in which no chordless cycle
  of length at least 4 contains a directed P3.
• These graphs are also perfect.

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Quasi-parity graphs

• A graph G is called quasi-parity (QP) if, for
  every induced subgraph H of G on at least two
  vertices, either H or its complement has an even
  pair.
• These are perfect.
• Bull-free graph is QP but not SQP.
• Graphs with no P5 or K5 is QP but not SQP.
• MAKE FIGURE OF GRAPH THAT ARE NOT QUASI-PARITY

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Perfect graphs that are not quasi-parity
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Strictly quasi-parity graphs

• A graph G is called strict quasi-parity (SQP) if
  every induced subgraph H of G either H is a
  clique or has an even pair.
• Every SQP graph is a QP graph. Converse not true.
• Every perfectly contractile graph is SQP.

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• Slim graphs, slender graphs, skeletal graphs are
  SQP.
• Oppositions graphs are SQP.
• Wing-triangulated graphs are SQP.

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Odd Pairs

• Two non-adjacent vertices x, y of a graph form an
  odd pair if all induced paths between x and y
  have an odd number
• of edges.
• No minimal imperfect graph contains an odd pair.

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Recent Progress

• A graph G with a clique-cutset C is perfectly
  contractile if and only if all the induced
  subgraphs GBi U C are perfectly contractile.
• Planer Graphs A planer graph with no odd hole
  and no odd stretcher is perfectly contractile.
• Claw-free graphs
• A graph is claw-free iff it does not contain
  the complete bipartite graph K (1,3) (known as
  the "claw") as an induced subgraph.

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• The line graph of any graph is claw-free.
• Every claw-free graph that contains no odd
  hole,no antihole,and no odd stretcher is
  perfectly contractile.

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• Bull -free graphs
• A bull is a graph consisting of a triangle and
  two pendant edges.
• A graph is said to be bull-free if it contains no
  induced subgraph isomorphic to a bull.
• Every bull-free Berge graph with no antihole is
  perfectly contractile.

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• Diamond-free graphs
• Any diamond free graph containing no odd hole
  and no stretcher is perfectly contractile.

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Bull Claw Diamond
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• Thank You