Title: Graph Partitions
1Graph Partitions
2-
- Partition V(G) into k sets
(k3)
3This kind of circle depicts an arbitrary set
This kind of line means there may be edges
between the two sets
4Special properties of partitions
- Sets may be required to be independent
5This kind of circle depicts an independent set
6This is just a k-colouring
(k3)
7Deciding if a k-colouring exists is
- in P for k 1, 2
- ? NP-complete for all other k
-
(k2)
8Deciding if a 2-colouring exists
1
Obvious algorithm
2
2
2
1
1
1
1
2
(k2)
9Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
10Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
11Deciding if a 2-colouring exists
Algorithm succeeds
2-colouring exists
No odd cycles
12G has a 2-colouring(is bipartite)
- if and only if it contains no induced
7 . . .
3
5
13Special properties of partitions
- ? Sets may be required to have no edges joining
them
14This kind of dotted line means there are no
edges joining the two sets
15- This is (corresponds to) a homomorphism.
- Here a homomorphism to C5 - also known
- as a C5-colouring.
16A homomorphism of G to H(or an H-colouring of G)
- is a mapping f V(G) ??V(H) such that
- uv ? E(G) implies f(u)f(v) ? E(H).
- A homomorphism f of G to C5 corresponds to
- a partition of V(G) into five independent sets
- with the right connections.
17f -1(1)
f -1(2)
f -1(3)
f -1(5)
f -1(4)
1
2
3
5
C5
4
181
2
3
5
4
19Special properties of partitions
- Sets may be required to be cliques
20This kind of circle depicts a clique
21- This is just a colouring of the
- complement of G
22G is a split graph
- if it is partitionable as
23Deciding if G is a split graph
24- G is split graph
- if and only if
- it contains no induced
5
4
25 Deciding if G is split
Algorithm succeeds
A splitting exists
No forbidden subgraphs
H-Klein-Nogueira-Protti
26This is a clique cutset
- (assuming all parts are nonempty)
27Deciding if G has a clique cutset
- is in P
- has applications in solving optimization problems
on chordal graphs - Tarjan, Whitesides,
28G is a chordal graph
- if it contains no induced
6 . . .
4
5
29G is a chordal graph
- if it contains no induced
6 . . .
4
5
if and only if every induced subgraph is either a
clique or has a clique cutset Dirac
30G is a cograph
- if it contains no induced
31G is a cograph
- if it contains no induced
if and only if every induced subgraph is
partitionable as or
Seinsche
32This kind of line means all possible edges are
present
33A homogeneous set (module)
- Another well-known kind of partition
34A homogeneous set (module)
- finding one is in P
- has applications in decomposition and recognition
of comparability graphs (and in solving
optimization problems on comparability graphs) - Gallai
35G is a perfect graph
? ?
- holds for G and all its induced subgraphs.
36G is a perfect graph
? ?
- holds for G and all its induced subgraphs.
- G is perfect if and only if G and its complement
- contain no induced
7 . . .
3
5
Chudnovsky, Robertson, Seymour, Thomas
37Perfect graphs
- contain bipartite graphs, line graphs of
bipartite graphs, split graphs, chordal graphs,
cographs, comparability graphs - and their complements, and
- model many max-min relations.
- Berge
38Perfect graphs
- contain bipartite graphs, line graphs of
bipartite graphs, split graphs, chordal graphs,
cographs, comparability graphs - and their complements, and
- model many max-min relations.
Basic graphs
39G is perfect
- if and only if it is basic or it admits a
- partition
- all others
-
Chudnovsky, Robertson, Seymour, Thomas
40Special properties of partitions
- Sets may be required to be
- Independent sets
- cliques
- or unrestricted
- Between the sets we may require
- no edges
- all edges
- or no restriction
41The matrix M of a partition
- 0 if Vi is independent
- M(i,i) 1 if Vi is a clique
- if Vi is unrestricted
- 0 if Vi and Vj are not joined
- M(i,j) 1 if Vi and Vj are fully joined
- if Vi to Vj is unrestricted
42The problem PART(M)
- Instance A graph G
- Question Does G admit a partition according to
the matrix M ?
43The problem SPART(M)
- Instance A graph G
- Question Does G admit a
- surjective partition according to M ?
- (the parts are non-empty)
44The problem LPART(M)
- Instance A graph G, with lists
- Question Does G admit a
- list partition according to M ?
- (each vertex is placed to a set on its list)
45For PART(M) we assume
- NO DIAGONAL ASTERISKS
- M has a diagonal of k zeros and l ones
- ( k l n )
46Small matrices M
- When M 4 PART(M) classified as being in P or
NP-complete - Feder-H-Klein-Motwani
- When M 4 SPART(M) classified as being in P
or NP-complete - deFigueiredo-Klein-Gravier-Dantas
- except for one matrix M
47Small matrices M with lists
- When M 4 LPART(M) classified as being in P
or NP-complete, except for one matrix - Feder-H-Klein-Motwani
- de Figueiredo-Klein-Kohayakawa-Reed
- Cameron-Eschen-Hoang-Sritharan
- When M 3 digraph partition problems
classified as being in P or NP-complete - Feder-H-Nally
48Classified PART(M)
- M has no 1s (or no 0s)
- H-Nesetril, Feder-H-Huang
49Classified PART(M)
- M has no 1s (or no 0s)
- H-Nesetril, Feder-H-Huang
- PART(M) is in P if M corresponds to a graph
- which has a loop or is bipartite, and it is
- NP-complete otherwise
- LPART(M) is in P if M corresponds to a bi-arc
- graph, and it is NP-complete otherwise
50Bi-Arc Graphs
- Defined as (complements of) certain intersection
graphs - A common generalization of interval graphs (with
loops) and (complements of) circular arc graphs
of clique covering number two (no loops).
51Classified PART(M)
- M has no 1s (or no 0s)
- H-Nesetril, Feder-H-Huang
- M has no s
-
52Classified PART(M)
- M has no 1s (or no 0s)
- H-Nesetril, Feder-H-Huang
- M has no s
-
- All PART(M) and LPART(M) in P Feder-H
53CSP(H)
- Given a structure T with vertices V(H) and
relations R1(H), Rk(H) of arities r1, , rk - Decide whether or not an input structure G with
vertices V(G) and relations R1(G), Rk(G), of
the same arities r1, , rk admits a homomorphism
f of G to H. - DICHOTOMY CONJECTURE Feder-Vardi
- Each CSP(H) is in P or is NP-complete
54Can all PART(M) be classified?
- If for every matrix M the problem PART(M) is
- in P or is NP-complete, then the Dichotomy
- Conjecture is true.
- Feder-H
- Thus hoping to classify all problems PART(M)
- appears to be overly ambitious
55G is complete bipartite
- if and only if it contains no induced
56G is a split graph
- if and only if
- it contains no induced
5
4
57G is a bipartite graph
- if and only if
- It contains no induced
7 . . .
3
5
58Another classification of PART(M)?
- For which matrices M can the problem
- PART(M) be described by finitely many
- forbidden induced subgraphs?
59Infinitely many forbidden induced subgraphs occur
- whenever M contains
- or
- Feder-H-Xie
60Do all others have finite sets of forbidden
induced subgraphs?
k
l
61Do all others have finite sets of forbidden
induced subgraphs?
62For small matrices M
- If M 5, all other partition problems have
only finitely many forbidden induced subgraphs
- If M 6, there are other partition problems
that have infinitely many forbidden induced
subgraphs -
- Feder-H-Xie
63means without
- If M 5, all other partition problems have
only finitely many forbidden induced subgraphs
- If M 6, there are other partition problems
that have infinitely many forbidden induced
subgraphs -
- Feder-H-Xie
64Restrictions to inputs G
- Since these partitions relate closely to
- perfect graphs, we may want to restrict
- attention to (classes of) perfect graphs G
65If M is normal
-
- The problem PART(M) restricted to
- perfect graphs G is in P
- Feder-H
- (fmfs)
66BUT
- classifying PART(M), for perfect G, as
- being in P or being NP-complete, would
- still solve the dichotomy conjecture
67If M is crossed
-
-
- The problem PART(M) restricted to
- chordal graphs G is in P
- Feder-H-Klein-Nogueira-Protti
68BUT
-
- there are problems PART(M), restricted to
- chordal graphs G, which are NP-complete
- Feder-H-Klein-Nogueira-Protti
69For all M
-
- A cograph G has a partition if and
- only if G does not contain one
- of a finite set of forbidden induced subgraphs
- Feder-H-Hochstadter
70Are these problems CSPs?
- Yes - two adjacent vertices of G have certain
allowed images in H and two nonadjacent vertices
of G have certain allowed images in H. (Two
binary relations)
71Are these problems CSPs?
- Yes - two adjacent vertices of G have certain
allowed images in H and two nonadjacent vertices
of G have certain allowed images in H. (Two
binary relations) - No - this is not a CSP(T), as inputs are
restricted to have each pair of distinct
variables in a unique binary relation.
72Full CSPs
- Given a set L of positive integers, an L-full
structure G has each k ? L elements in a unique
k-ary relation - CSPL(H) is CSP(H) restricted to L-full structures
G
73Example with m binary relations
- Given a complete graph with edges coloured by
- 1, 2, , m.
- Given such a G, colour the vertices 1, 2, ,
m, - without a monochromatic edge
?
i
i
i
74- When m 2, the problem is in P
75- When m 2, the problem is in P
- When m ? 4, it is NP-complete
76- When m 2, the problem is in P
- When m ? 4, it is NP-complete
- When m 3, we only have algorithms of complexity
- n O ( log n / log log n ) FHKS
77- An algorithm of complexity nO(log n) solving
the (more general) problem with lists - Given a complete graph G with
- edges coloured by 1, 2, 3, and
- vertices equipped with lists ? 1,2,3
78If all lists have size ? 2
- Introduce a boolean variable for each vertex (use
the first/second member of its list) - Express each edge-constraint as a clause of two
variables - Solve by 2-SAT
79In general
- Let X be the set of vertices with lists 1,2,3
- Recursively reduce X as follows
- Try to colour G without giving any vertex its
majority colour - Give each vertex in turn its majority colour
X
? (X-1) / 3
X
X
80Analysis of Recursive Algorithm
- Time to solve problem with X x
- T(x) (1 x T(2x/3)) . T(2-SAT)
81Analysis of Recursive Algorithm
- Time to solve problem with X x
- T(x) (1 x T(2x/3)) . T(2-SAT)
- ? T(x) x O(log x)
82Analysis of Recursive Algorithm
- Time to solve problem with X x
- T(x) (1 x T(2x/3)) . T(2-SAT)
- ? T(x) x O(log x)
- ? T(n) n O(log n)
83Can we say anything ?
- A kind of (quasi) dichotomy
- If 1 ? L then every CSPL(H) is
- ? quasi-polynomial or
- ? NP-complete
- Feder-H