Subgraphs

1
Subgraphs

• Lecture 4

2
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.

3
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.

4
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.

5
Subgraphs - Example

• 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.

a
1
2
c
b
d
e
3
4
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Subgraph Types

• Open subgraph
• Induced subgraph
• Spanning subgraph
• Isometric subgraph
• Convex subgraph

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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.

8
Trivial Subgraph

• Subgraph H is trivial, if either H f, or H G.

9
Exercise

• N7. Prove that G is connected if and only if it
  has not nontrivial open subgraphs.

10
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.

11
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.

12
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 ¹
  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.

13
k-connectedness

• Graph G with V(G) gt k is k-connected, if a
  removal of any set S with S lt 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) gt W(G ).
• A connected graph with no cut-vertex is called a
  block.

14
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.

15
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.

16
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.

17
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.

18
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.

19
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!

20
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.

21
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 ltUgt or, if we wnat to refer to the
  original graph by GU.
• Example P5 is an induced subgraph of C6.

22
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 ltV1(v)gt 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.

23
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).

24
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).

25
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.

26
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.

27
Exercises

• N25. For H µ G define the convex 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) ¹ 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 is a
  path.
• N30. Characterize graphs, with the property that
  each interval is a path.

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28
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.