Transcript Subgraphs Lecture 4

Subgraphs
Lecture 4
Bipartite Graphs
• A graph is bipartite, if
the vertex set can be
partitioned into two
bipartitions, say G and
R, such that each edge
has one endpoint in G
and the ogther in R.
• Graph on the left is
biparitite.
Exercises
• N1: Show that each Km,n. is bipartite.
• N2: Show that each Qn is bipartite.
• N3(*): Show that a graph is bipartite if and only if
it has no odd cycles.
• N4: Which generalized Petersen graphs G(n,k) are
bipartite?
• N5: Prove that each tree is a bipartite graph.
• N6: Prove that X is bipartite, if and only if each of
its components is bipartite.
Subgraphs
• Graph H=(U,F) is subgraph of graph
G=(V,E), if U µ V and F µ E.
• Warning! It is important that (U,F) is
indeed a graph! For each edge from F must
have both of its endpoints in U.
Subgraphs - Example
a
1
2
c
d
b
e
3
4
• G=(V,E)
• VG ={1,2,3,4}
• EG = {a,b,c,d,e}
Let: U = {1,2,3}, W =
{2,3,4}, F = {b}, P =
{a,d}. Then (U,P) and
(W,F) are subgraphs
while (U,F) and (W,P)
are not.
Subgraph Types
•
•
•
•
•
Open subgraph
Induced subgraph
Spanning subgraph
Isometric subgraph
Convex subgraph
Open Subgraph
• Subgraph H=(U,F) of graph G=(V,E) is
open, if each ede e 2 E has either both
endpoints in U, or none.
Trivial Subgraph
• Subgraph H is trivial, if either H = f, or H
= G.
Exercise
• N7. Prove that G is connected if and only if
it has not nontrivial open subgraphs.
Connected Component
• Minimal nontrivial open subgraph is called
a connected component of G. By W(G) we
denote the number of connected
components of graph G.
Distance in Connected Graph
• Each connected graph G gives rise to a
metric space (V,dG) for dG(u,v) being the
length of shortest path in G, from u to v.
Distance Partition
• For a given graph G and a given vertex v we may
define the k-th link: Vk := {u 2 V(G)| d(v,u) = k}.
• This defines a partiton V = {V0,V1,...,Ve} , Vk  ;
of the vertex set V(G) = V0 t V1 t ... t Ve. The
number e is called the excentricity of vertex v.
Maximum excentricity is called the diameter of
graph.
• This partition is called the distance partition of G
with respect to v.
• Clearly, V0 = {v}.
k-connectedness
• Graph G with |V(G)| > k is k-connected, if a
removal of any set S with |S| < k stays conneced.
• Connectivity k(G) of graph G is the largest k,
such that G is still k-connected.
• Vertex v of graph G is a cut-vertex, if W(G – v)
> W(G ).
• A connected graph with no cut-vertex is called a
block.
2-connectedness
• Theorem: The following claims are
equivalent:
– Graph G is 2-connected,
– Graph G is a block,
– Any pair of vertices belongs to a common
cycle.
Menger Theorem
• Two paths in a graph with common
begining vertex and a common end-vertex
are internally disjoint, if they have no
other vertex in common.
• Theorem: Graph is k-connected, if and only
if there are k pair-wise internally disjoint
paths between any two of its vertices.
Spanning Subgraph
• If H=(U,F) is a subgraph of G(V,E) and U =
V, then H is called a spanning subgraph of
G.
Spanning Paths and Cycles
• A spanning subgraph is also called a factor.
• A spanning path in a graph is also called a
hamilton path.
• A spanning cycle in a graph is also called a
hamilton cycle.
Spanning Trees
• Each connected graph has a spanning
tree.
• For finite graphs the proof is not hard. As
long as we do not get a tree we remove
edges from any cycle.
• For infinite graphs this fact is equivalent to
the axiom of choice.
How many spanning trees does
the complete graph have?
• On the right K3 has
three spanning trees!
• Let t(G) denote the
number of spanning
trees in G.
• Theorem: t(Kn) = nn-2
• Proof: Prüfer code!
Exercises
• N8. Show that if G has a hamilton cycle it also
contains a hamilton path.
• N9. Show that every graph that has a hamilton
path is connected..
• N10. Construct a graph on 10 vertices that has no
hamilton path.
• N11. Construct a graph on 10 vertices that has no
hamiloton cycle but has a hamilton path.
• N12: Construct a graph on 10 vertices that has a
hamilton cycle.
Induced Subgraph
• Graph H is an induced subgraph
of graph G, if H is obtained
from G by removing the
vertices from V(G)-V(H).
• An induced subgraph of G is
determined by its vertrex set U
µ V(G). If we want to
distinguish the graph from its
vertex set we denote the former
by <U> or, if we wnat to refer
to the original graph by G|U.
• Example: P5 is an induced
subgraph of C6.
Exercises
• N13. Prove the following: In a connected graph G there
exsists at least one distance partition such that each k-link
Vk is an independent set if and only if G is bipartite.
• N14. Let G and H be graphs. We say, that G is locally H if
and only if for each vertex v 2 V(G) the first link <V1(v)>
is isomorphic to H. Find a graph that is locally P3.
• N15. Prove that K2,2,2 is locally C4.
• N16. Determine all graphs with diameter 1.
• N17. Use the result of N13 to show that if one distance
partion has independent k-links then all of them have
independent k-links.
• N18. Use N17 to design an algorithm that will find a
bipartition of a bipartite connected graph.
Isometric Subgraph
• H=(U,F) is an isometric subgraph of graph
G=(V,E), if the distances are preserved:
• For each u,v 2 U: dH(u,v) = dG(u,v).
Interval IG(u,v)
• Let u, v 2 V(G) belonging to the same
connected component of G. By IG(u,v) we
denote the interval with endpoints u and v.
• IG(u,v) is the graph, induced on the set of
vertices belonging to some shortest path
from u to v.
• If there is no danger of confusion wecan
simplify notation: I(u,v).
Convex Subgraph
• Graph H is a convex subgraph of G, if for
every pair of vertices u and v from the
V(H) that belong to the same connected
component of G, the interval IG(u,v) is a
subgraph of H.
Exercises
• N19. Prove that each convex subgraph is an isometric subgraph.
• N20. Prove that each isometric subgraph is an induced
subgraph.
• N21. Prove that each connected component is a convex
subgraph.
• N22. Prove that the intersection of two induced subgraphs is an
induced subgraph..
• N23. Prove that the intersection of two convex subgraphs is a
convex subgraph..
• N24. Determine all intervals of the cube Q3.
Exercises
7
6
8
1
2
• N25. For H µ G define the convex
5
closure cvx(H) of H in G. Compute
cvx(Pk) in Cn.
• N26. Prove that each interval I(a,b) is
a subgraph of cvx(a,b).
• N27. Determine all intervals in the
graph G on the left. Find two vertices
a and b of G that have I(a,b) 
4
cvx(a,b).
• N28. Prove that althouth the subgraph
induced by any shortest path in G is
isometric, there are intervals that are
not isometric subgraphs.
• N29. Prove that each interval in a tree
3
is a path.
• N30. Characterize graphs, with the
property that each interval is a path.
Homework
• H1. Let C be the shortest cycle in graph G.
Show that C is an induced subgraph of G.
• H2. Determine all non-isomorphic intervals
in Q4.
• H3. Find an isometric subgraph of Q3 that is
not convex.