Seminario Avanzado de Teoría de Grafos

Seminario Avanzado de Teorı́a de Grafos
Flavia Bonomo
DC, FCEyN, Universidad de Buenos Aires, Argentina
2016
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
1/9
Outline of the course: algorithmic problems
Maximum clique
Maximum stable set
Vertex coloring
Minimum clique cover
Minimum dominating set
(All NP-complete in general.)
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
2/9
Outline of the course: graph classes and concepts
modular-decomposition
perfect
circular-arc
circle
cographs
chordal
permutation
interval
block
trees
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
3/9
Desirable properties for a graph class C
Hereditary: G ∈ C and H induced subgraph of G then H ∈ C !
There exists a characterization by minimal forbidden induced
subgraphs, even if the family is infinite.
Hereditary on subgraphs: G ∈ C and H subgraph of G then H ∈ C.
(Sometimes it is too much to require, every graph is a subgraph of a
large enough clique.)
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
4/9
Desirable properties for a graph parameter ϕ
Monotonicity: H induced subgraph of G then ϕ(H) ≤ ϕ(G ) (or
ϕ(H) ≥ ϕ(G ), depending on the problem).
Monotonicity on subgraphs: H subgraph of G then ϕ(H) ≤ ϕ(G ) (or
ϕ(H) ≥ ϕ(G )).
Monotonicity on spanning subgraphs: H spanning subgraph of G
(V (H) = V (G )) then ϕ(H) ≤ ϕ(G ) (or ϕ(H) ≥ ϕ(G )).
Problem
Monotonic
M. on subgraphs
M. on spanning s.
clique
≤
≤
≤
stable set
≤
x
≥
coloring
≤
≤
≤
clique cover
≤
x
≥
dominating set
x
x
≥
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
5/9
Connectivity
The property of being connected is not hereditary (hereditary version:
cliques).
Cannot be expressed by minimal forbidden induced subgraphs.
Most of the problems (specially monotonic parameters) can be solved
on each component.
We will often assume connection but we will not require it.
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
6/9
Intersection graphs
Trees and block graphs
The very basics: trees and forests
G is a forest iff it has no cycles Cn , n ≥ 3. ! characterization by
forbidden induced subgraphs, indeed also by forbidden subgraphs
a tree is a connected forest
every non-trivial tree has at least two leaves ! dismantling/building
sequence
every vertex v that is not a leaf is a cutpoint and decomposes the
tree into d(v ) parts ! decomposition/composition theorem
every edge is a bridge ! decomposition/composition theorem
for every vertex v and every k, Nk (v ) is a stable set ! level structure
a tree is bipartite ! vertex partition
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
7/9
Intersection graphs
Trees and block graphs
The very basics: trees and forests
Algorithmic use:
List-coloring ! leaves + recursion
Maximum stable set ! cutpoint/level structure + dynamic
programming
Minimum dominating set ! cutpoint/level structure + dynamic
programming
Minimum clique-covering ! forced using leaves
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
8/9
Intersection graphs
Trees and block graphs
How to generalize trees? Block graphs
The property of cutpoints in trees is equivalent to every biconnected
component (block) is a vertex or an edge.
Block graph: every biconnected component (block) is a clique.
Forbidden induced subgraphs: Cn , n ≥ 4, diamond.
leaves ! end blocks
Many algorithms for trees extend to block graphs.
Diamond-free ⇔ every edge belongs to one maximal clique.
diamond
Flavia Bonomo (DC–FCEN–UBA)
SATG
2016
9/9