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

6. Subgraph Types Open subgraph Induced subgraph Spanning subgraph
Isometric subgraph
Convex subgraph

7. 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)| > 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.

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 <U> or, if we wnat to refer to the original graph by G|U.
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 <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.

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. 6 5 7 8 4 2 3 1

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.