Subgraphs Lecture 4 Bipartite Graphs A graph is

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 • N 1: Show that each Km, n. is bipartite. • N 2: Show that each Qn is bipartite. • N 3(*): Show that a graph is bipartite if and only if it has no odd cycles. • N 4: Which generalized Petersen graphs G(n, k) are bipartite? • N 5: Prove that each tree is a bipartite graph. • N 6: 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 1 a 2 c d 3 e b 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 • N 7. 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, d. G) for d. G(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 = {V 0, V 1, . . . , Ve} , Vk ¹ ; of the vertex set V(G) = V 0 t V 1 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, V 0 = {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 K 3 has three spanning trees! • Let t(G) denote the number of spanning trees in G. • Theorem: t(Kn) = nn-2 • Proof: Prüfer code!

Exercises • N 8. Show that if G has a hamilton cycle it also contains a hamilton path. • N 9. Show that every graph that has a hamilton path is connected. . • N 10. Construct a graph on 10 vertices that has no hamilton path. • N 11. Construct a graph on 10 vertices that has no hamiloton cycle but has a hamilton path. • N 12: 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: P 5 is an induced subgraph of C 6.

Exercises • N 13. 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. • N 14. 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 <V 1(v)> is isomorphic to H. Find a graph that is locally P 3. • N 15. Prove that K 2, 2, 2 is locally C 4. • N 16. Determine all graphs with diameter 1. • N 17. Use the result of N 13 to show that if one distance partion has independent k-links then all of them have independent k-links. • N 18. Use N 17 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: d. H(u, v) = d. G(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 • N 19. Prove that each convex subgraph is an isometric subgraph. • N 20. Prove that each isometric subgraph is an induced subgraph. • N 21. Prove that each connected component is a convex subgraph. • N 22. Prove that the intersection of two induced subgraphs is an induced subgraph. . • N 23. Prove that the intersection of two convex subgraphs is a convex subgraph. . • N 24. Determine all intervals of the cube Q 3.

Exercises 7 6 8 1 2 • N 25. For H µ G define the convex 5 closure cvx(H) of H in G. Compute cvx(Pk) in Cn. • N 26. Prove that each interval I(a, b) is a subgraph of cvx(a, b). • N 27. 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). 4 • N 28. Prove that althouth the subgraph induced by any shortest path in G is isometric, there are intervals that are not isometric subgraphs. • N 29. Prove that each interval in a tree is a path. 3 • N 30. Characterize graphs, with the property that each interval is a path.

Homework • H 1. Let C be the shortest cycle in graph G. Show that C is an induced subgraph of G. • H 2. Determine all non-isomorphic intervals in Q 4. • H 3. Find an isometric subgraph of Q 3 that is not convex.