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

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} , Vk¹; of the vertex set V(G) = V0t V1t ... 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. • Connectivityk(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 6 5 7 • 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. 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.