Path kernels and partitions

1
Path kernels and partitions

• Peter Katrenic
• Institute of Mathematics
• Faculty of Science
• P. J. Šafárik University, Košice

2
Coloring

• G(V,E)

?(G)kl
?V1,V2 V1?V2? V1?V2 V ?(GV1)k ?(GV2)l
V1
V2
k2l2
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Graph decomposition with bounded maximal degree

• G(V,E)

?(G)kl
?V1,V2 V1?V2? V1?V2 V ?(GV1)k ?
(GV2)l
V1
V2
k2l3
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Path partition conjecture

• G(V,E)

?(G)kl
?V1,V2 V1?V2? V1?V2 V ? (GV1)k ?
(GV2)l
V1
V2
k3l5
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Pn-kernel and Pn-semikernel

• A subset K of V(G) is called Pn-kernel of G
  if
• ?(GK) n-1
• every vertex v ?V(G-K) is adjacent to a
  Pn-1-terminal vertex of GK.

• A subset S of V(G) is called a Pn-semikernel of G
  if
• ?(GS) n-1
• every vertex in N(S)-S is adjacent to a
  Pn-1-terminal vertex of GS.

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Difference between kernel and semikernel
x6
x7
x4
x2
x1
x3
x8
x5
Mx2,x3,x4,x5
No
-is M P5-kernel in G?
Yes
-is M P5-semikernel in G?
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Conjecture about kernels
G(V,E)
For every n2 exists K?V, that K is Pn-kernel
in G.
n4
8
Conjecture about semikernels
G(V,E)
For every n2 exists S?V, thatS is
Pn-semikernel in G.
n4
9
Conjectures
Every graphis ?-partitionable
Every graph has Pn-kernelfor every n2
Every graph has Pn-semikernel for every n2
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Relationship among path kernels, semikernels and
partitions

• Let P be a hereditary class of graphs. If every
  graph in P has Pn-semikernel, then every graph in
  P has Pn-kernel.
• Let G be a graph with ?(G)ab, ab. If G has
  Pb1-semikernel, then G is (a,b)-partitionable.

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The existence of Pn-kernels for small values of n

• Dunbar,Frick
• Every graph has P7-kernel
• Melnikov,Petrenko
• Every graph has P8-kernel
• P.K.(2005)
• Every graph has P9-kernel

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Graphs that have Pn-kernels for all n

• If every block of graph G is either a complete
  graph or a cycle, then G has Pn-kernel for all
  n2.
• Let G be a complete multipartite graph. The G has
  Pn-kernel for all n2.

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Cycle lengths and path kernels

• If G is graph with g(G)n-2, then G has Pn-kernel
• Let G be a graph with ?(G)ab, ab. If g(G)a-1,
  then G is (a,b)-partitionable
• Let G be a graph with ?(G)ab, ab. If c(G)a1,
  then G is (a,b)-partitionable

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Absent of Pn-kernels for big values of n

• Thomassen
• Exists graph, that dont have a P364-kernel
• P.K.(2005)
• For every n364 exists graph, that dont have
  a Pn-kernel

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Algorithm to find Pn-semikernel for small values
of n

• Let SHi, where i is smallest integer such that
  Hi is a subgraph of G.
• Algorithm
• Initially we let BV(G)-S and A?.
• Identify all P8 terminal vertices of S and move
  all their B-neighbours to A. If N(S) n B is
  empty, then stop, else 2.
• If two vertices x and y in S have a common
  B-neighbour, then move one common B-neighbour of
  x and y to S and return to 1.
• If some P7-terminal vertex x of S has a
  B-neighbour, then move one B-neighbour of x to S
  and return to 1, else 4.
• 5 6
• If some P3-terminal vertex x of S has a
  B-neighbour, then move one B-neighbour of x to S
  and return to 1, else 2.

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Sequence of graphs Hi for n7
H1
H2
H3
H4
H5
Príklad
H6
H7
H8
H9
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Summary of conjecture status

• Conjuncture about kernels is true for every n 9
  a is false for every n 360
• If G is graph with ?(G)17, then G is
• ?-partitionable

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• Thanks for your attention.