1. Advanced Operations on Graphs Lecture 6

2. Cone and Suspension The join of G and K 1 we call the cone over G and is denoted by Cone(G) = G*K 1. The join G*(2K 1 ) is called suspension.

3. Examples Any complete multipartite graph is a join of empty graphs. The cone Cone(C n ) is called a pyramid or wheel W n. Octahedral graph is the suspenstion over C 4. It can be written in the form: –O 3 = (2K 1 )*(2K 1 )*(2K 1 ). Construction can be generalized to: –O n = (2K 1 )*(2K 1 )*...*(2K 1 )

4. Exercises N1: Prove that for any graph X at least one of the graphs X and X c is connected. N2: Decribe two graphs G and H, so that H is isomorphic to an induced subgraph, and also to a non-induced subgraph of G. N3: Graph of our original example is isomorphic to K 4 -e. 1.How many subgraphs has K 4 -e? 2.How many non-isomorphic subgraphs has K 4 -e? 3.How many induced subgraphs has K 4 -e? 4.How many non-isomorphic induced subgraphs has K 4 -e?

5. Exercises N4. Determine the number of vertices and egdes of the generalized octahedral graph O n. N5. Let V = {-1,1} n. Define a graph G n, whose vertex set is V and two vertices are adjacent if and only if d(u,v) 2 < n. Prove that G n is isomorphic to O n. N6. Explore the relationship between graphs G 1 = G * (2K 1 ) and G 2 = ((G*K 1 )*K 1 ). N7. True or False? The Cone(H) is convex in Cone(G) if H is convex in G. N8. Show that G\H is a spanning subgraph of G.

6. Homework H1. Prove that C 8 is isomorphic to its bipartite complement C 8 b. H2. Determine all paths P n that are isomorphic to their bipartite complements P n b. H3. Draw suspension over C 5. H4. Graph, that is isomorphic to its complement is called self-complementary. Prove that there exists no self- complementary graph on (n+2) vertices, if there exist self- complementary graphs on n and on (n+1)-vertices. H5. Draw all self-complementary paths and all self- complementary cycles.

7. Subdivision Let e 2 E(G) be an edge of G. Let S(G,e) denote the graph obtained from G by replacing the edge e by a path of length 2 passing through a new vertex.. Such an operation is called subdivision of the edge e.. Let F be a subset of E(G), then S(G,F) denotes the graph obtained from the subdivision of each edge of F. In the case F = E, we drop the second argument and S(G) denotes the subdivision graph G. Graph H is a general subdivision of graph G, if H is obtained from G by a sequence of edge subdivisions..

8. Graph Homeomorphism Graphs G and H are homeomorphic, if they have a common subdivision. Graph G is topologically contained in a graph K, if there exists a subgraph H of K, that is homeomorphic to G.

9. Exercises N1: Prove that S(G) is bipartite for any simple G. N2: Find a general subdivision of K 3,3 in G(5,2). N3: Given a graph X with n vertices and m edges. Determine the number of verticexs and the number of edges of S(X)? N4: Graph G is topologically almost contained in graph K, if there exists a subgraph H of K, that is a general subdivision of G. Prove that topological containment implies topological almost containment and find a counterexample for the converse.

10. Matching Edges with no common endvertex are called independent. A set of pairwise independent edges is called a matching.

11. Maximal Matching A matching that cannot be augmented by adding new edges is called a maximal matching.

12. Perfect Matching Proposition: Let M be a matching of a graph G on n vertices. Then |M| · n/2. A matching M with |M| = n/2 is called a perfect matching.

13. Exercises N5. Prove that |M| · n/2 for any matching. N6. Prove that if a graph has a perfect matching then it has an even number of vertices. N7. Prove that a parfect matching is maximal matching. Give an example of a maximal matching that is not perfect. N8. Show that the family of matchings define an abstract simplicial complex on E(G).

14. Abstract Simplicial Complex K µ P (S) is an abstract simplicial complex if for each  2 K and each  µ  it follows that  2 K. On the left: K = { ;, a, b, c, d, e, f, g, h, ab, ad, abd, bc, be, bcd, bd, ce, df, dg, de, eh} a e d c b h g f

15. Exercises N9: Find a perfect matchning in Petersen graph G(5,2). N10: Find a cubic graph with no perfect matching.

16. Line Graph L(G) Two edges with a common end-vertex are incident. Incidence is a binary relation on the edge set E(G). Line graph L(G) has the vertex set E(G), while the edges of L(G) are determined by the incidence of edges in G.

17. Examples Top row depicts the Heawood graph and its fourvalent linegraph. Bottom row depicts the Petersen graph and its line graph.

18. Exercises N11: Show that the line graph of a regular graph is regular. N12: What is then number of vertices and the number of edges of L(G), given the number of vertices n and the number of edges m in G? N13: Define trunacated graph T(G) := L(S(G)). Determine the number of vertices and edges in T(G)? N14: Draw T(K 4 ) and T(Q 3 ). N15: Define operations L(G) and T(G) only partially, for some set of vertices or edges, similarly as S(G,F) was defined. For instance, for bipartite graphs we may define operations with respect to only one bipartition set.

19. Variations of Graphs Our main topic are simple finite graphs. We have to be aware of other similar creatures. Infinite graphs Digraphs General graphs (multigraphs) Pregraphs Rooted graphs...

20. Infinite Graphs An infinite graph may have an infinite set of vertices or edges. Countable graphs have both sets coutable (or finite) Among countable graphs most tractable are locally finite graphs. (Each vertex has finite valence). Even more restricted are bouded valence graphs, where  (G) is finite, in particular regular ones.

21. Examples Examples of infinite graphs: Each metric space (X,d) determines a unit distance graph: »V(X) := X »x ~ y, d(x,y) = 1. Ray or infinite path P 1 Double ray or infinite cycle C 1 Infinite k-way tree T( 1,k). Infinite square lattice Q 1. Infinite triangular lattice T 1. Infinite hexagonal lattice H 1.

22. Digraphs In case of directed graphs (digraphs) there are no undirected edges e = uv, but there are directed edges or arcs: a = (u,v). uv e = uv uv a = (u,v)

23. Digraphs from Binary Relations Let R µ V £ V be a binary realtion on V. R defines in a natural way a digraph.

24. Exercises N16: Following the definion of a category, define a digraph as a quadruple (V,A,i,t), where i and t are mappings, that assign to each arc a its intial vertex i(a) and terminal vertex t(a). Explain the procedure that starts with a binary relation R and obtains a digraph (V,A,i,t)? N17: Define homomorphisms and isomorphisms of digraphs. N18: Formally define a directed path P n !, directed cycle C n ! and directed complete graph K n !. N19: Describe a digraph that does not arise from a binary relation.. N20: Decribe a procedure that assings to each simple undirected graph (V,~) a directed graph in such a way that each undirected path is assigned a pair of oppositely directed arcs.

25. Loops, parallel edges, multigraphs A multigraph (general graph) is more general than a simple graph, since it may have parallel edges (between vertices u and v we may have two edges e = uv and f = uv) or loops. A loop z = uu is na edge that has both endpoints the same. C 2, parallel edges C 1, loop u u v

26. Dipoles, bouquets of circles and more Among general graphs there are several interesting families. A dipole  n has two vertices and n parallel edges between them. Bouqet of circles B n has one vertex with n loops. The handcuff graph G(1,1) is a graph with two vertices, each one having an edge and another edge joining them. 33 B4B4 The handcuff graph G(1,1)

27. Pregraphs If we want to distinguish a loop from a half- edge we need a notion of a pregraph.

28. Definition of pregraph Pregraph X = (V,S,i,r) consists of vertex set V, set of arcs S, mapping intial vertex, i: S ! V involution reverse r: S ! S. Since r is an involution, we have r 2 = 1. The orbits of r of length 2 are edges, while orbits of length 1 are halfedges. A pregraph without half-edges is a general graph.

29. Rooted Graphs Let X be a graph and let r 2 V(X) be a selected vertex. A pair (X, r) is called a rooted graph and x is called the root. The idea of root can be generalized to a set of vertices or even a subgraph Y µ X. In some computer systems the root graph Y corresponds to the selected part of graph.. The idea can be generalized to “graduated” graphs: Y 0 = r µ Y 1 µ... µ Y k = X.

30. Edge Contraction Let e = uv be an edge of G. By G/e we denote the graph obtained from G, by contraction of e. This means we identify vertices u and v, remove the loop and possible extra parallel edges.

31. Exercise N21: Prove that K 5 can be obtained from G(5,2) by a series of edge contractions.

32. Graph Products For two graphs G and H we may define several graphs on the vertex set V(G) £ V(H) that behawe like a product. The most natural one is the so-called Cartesian product that we introduce first.

33. Square Grid Gr(n,m). Let Gr(n,m) denote the graph defined by a square grid in the plane, determined by n  m nodes. We will call it a square grid. For instance, Gr(2,2) is isomorphic to C 4. We may regard Gr(n,m) as a product of paths P n and P m. In a similar way we may regards the n-prism graph as the product of a cycle C n and K 2. This is a motivation for the introduction of the Cartesian product of graphs.

34. Cartesian Product of Graphs Cartesian product G H of graphs G and H has the vertex set VG  VH. Two vertices (u,v) and (u’,v’) are adjacent if and only if: u = u’ and v ~ v’ or u ~ u’ and v = v’.

35. More products. Cartesian product is not the only product for graphs. If we define a category of graphs and maps that are dimension preserving the categorical product is the so-called tensor product. If, however, the category admits also graph mappings that may map edges to vertices, the catregorical prodiuct is the so-called strong product. We will meet both of them.

36. Tensor Product of Graphs Tensor product G  H of graphs G and H has the vertex set VG  VH. Two vertices (u,v) and (u’,v’) are adjacent if and only if: u ~ u’ and v ~ v’.

37. Strong Product of Graphs Strong product G ⊠ H of graphs G and H has the vertex set VG  VH. Two vertices (u,v) and (u’,v’) are adjacent if and only if: –u ~ u’ and v ~ v’ or –u = u’ and v ~ v’ or –u ~ u’ and v = v’.

38. Examples of Products On the left, there are cartesian, tensor and strong products of P 5 by P 4. [Cartesian product is the grid.] P 5 ¤ P 4 P 5 ⊠ P 4 P 5 £ P 4

39. Exercises N22: Let the graph G i have n i vertices and m i edges, for i = 1,2. Determine the number of vertices and edges in their Cartesian product G 1 ¤ G 2. N23: Let the graph G i have n i vertices and m i edges, for i = 1,2. Determine the number of vertices and edges in their tensor product product G 1 £ G 2. N24: Let the graph G i have n i vertices and m i edges, for i = 1,2. Determine the number of vertices and edges in their strong product G 1 £ G 2. N25: Under what conditions will the Cartesian product G 1 ¤ G 2 be connected? N26: Under what conditions will the tensor product G 1 £ G 2 be connected? N27: Under what conditions will the strong product G 1 £ G 2 be connected? N28: Under what conditions will the Cartesian product G 1 ¤ G 2 be bipartite? N29: Under what conditions will the tensor product G 1 £ G 2 be bipartite? N30: Under what conditions will the strong product G 1 £ G 2 be bipartite?

40. Factors A spanning subgraph is also called a factor. A k- valent regular factor is simply called a k-factor. 1-factor is a different name for a perfect matching. A 2-factor is a disjoint union of cycles covering the vertex set of the graph.

41. Exercises N31. Prove: If a trivalent graph contains a 1-factor, it also has a 2-factor. N32. Find a trivalent graph, without a 1- factor.

42. Factorization Let G = (V,E) be a graph and let E be disjoint union of sets E = F 1 t F 2 t... t F s, then H i = (V,F i ) are factors of G and the decompostion of G into these factors is called a factorization. This is written as: G = H 1  H 2 ...  H s If all factors H i are k-factors, we speak of k- factorization of graph G.

43. Example. For any graph G and any of its spanning subgraphs H we have factorization: G = H  G\H.

44. Exercises N33: Prove that only regular d-valent graphs with d divisible by k admit a k-factorization. N34: Prove that non-regular graphs have no 1- factorizations. N35: Show that a 2-valent graph G has 1-factorziation, if and only if it is a disjoint union of even cycles. N36: Prove that each prism  n has a 1-factorization. N37. Prove that Petersen graph has no 1-factorization. N38. Show that each cubic hamiltonian graph has a 1- factorization.

45. Graph Power For a connected graph G and an integer k, we define the k-th graph power G (k) as follows: V(G (k) ) := V(G). u ~ v if and only if d(u,v) · k.

46. Pure Graph Power For a connected graph G and an integer k, we define the k-th pure graph power G [k] as follows: V(G [k] ) := V(G). u ~ v if and only if d(u,v) = k.

47. Exercises N39. Prove that for any integer k and graph connected G on n vertices we have: G (k) := G [0] = E n. G (k) = G [0] © G [1] ©... © G [k].

48. Intermezzo - Partitions, Set Partitions, Graudated sets, etc. Equivalence relation – Set Paritition Distance Partition – Ordered Set Parition Graduated Graph – Graduated Set Valence Sequence – (Number) Partition Hierarchy – Nested (Graduated) Set Partitions Rooted Tree, Rooted Graph... MINIVEGA should support all these structures.

49. Homework H1. Prove that each prism graph  n = K 2 ¤ C n admits a 1-factorization. H2. Prove that each of the three products (cartesian, tensor, strong) is associative and commutative (up to isomoprhism). H3. Give a definition of the following infinite graphs (a suitable drawing suffices): P 1, C 1, T( 1,k), Q 1,T 1, H 1,