-93- 11. Graph Theory Def: V:non-empty vertexornode set E ⊆ V ×V :edge set G ≡ (V, E):directed graph on V, or digraph on V Example: b e f V={a, b, c, d, e, f, g} a E={(a,b), (b,c), (c,a), ... } f c d Note: (a, b) ≠ (b, a) Def: When there is no concern about the direc- tion of an edge, the graph is called undirected. In this case, an edge between twovertices a and b is represented by {a, b}. Note: {a, b} = {b, a} -94- b . {a,b} a {a,a} {a, a}isaloop from a to a. If a graph is not specified as directed or undi- rected, it is assumed to be undirected. Def: G = (V, E): undirected graph An x − ywalk is a finite alternating sequence x = x0, e1, x1, e2,...,en−1, xn−1, en, xn = y of vertices and edges from G,starting at ver- tex x and ending at vertex y,with the n edges ei = {xi−1, xi}, 1 ≤ i ≤ n length = n =noofedges If n = 0, a trivial walk -95- When x = y and n >0,called a closed walk, otherwise, open walk. x x1 2 e2 e . e1 3 x = x0 x3 = y Def: If no edge in an walk is repeated, then the walk is called a trail.Aclosed trail is called a circuit. When no vertexofthe x − y walk occurs more than once, then the walk is called a path.Aclosed path is called a cycle. (The term "cycle" will always imply the pres- ence of at least 3 distinct edges) End . Start Trail Path -96- Fordirected graphs, we put "directed" in front of all the terms defined above. Theorem: G = (V, E): undirected graph a, b ∈ V, a ≠ b If there exists a trail from a to b then there is a path from a to b. Proof. Let T be the set of all trails from a to b. T has an element with the smallest length. Let P be such a trail. Then P must be a path. Why? . a b -97- Def: G = (V, E): undirected graph G is called connected if there is a path between anytwo distinct vertices of G. Connected Not connected Adigraph is said to be connected if the asso- ciated undirected graph is connected. Def: G = (V, E)(directed or undirected) Agraph G1 = (V1, E1)iscalled a subgraph of G if V1⊆V and E1⊆E (edges in E1 must be incident with vertices in V1) -98- Def: G = (V, E)(directed or undirected) Aconnected subgraph of G is said to be a component of G if it is not properly contained in anyconnected subgraph of G. Notation: The number of components of G is denoted by K(G). Note: Agraph is connected iff K(G) = 1 Def: = Ag_ raph G (V, E)iscalled a multigraph if _| a, b ∈ V, a ≠ b,such that there are twoor more edges between a and b (undirected) (from a to b (directed)). Def: multiplicity = 3 n −graph: no vertexhas multiplicity > n -99- Example: (No. 8, page 412) . Removing anyedge would result in a discon- nected graph Can such a graph be characterized using the concept of graph path? (the entire graph is a path) Example: (No. 9, page 412) If a graph satisfies the above condition then -itmust be loop-free - G can not be a multigraph -If G has n vertices then it must have n − 1 edges -100- Example: (No. 10, page 412) a) If G = (V, E): undirected, |V| = m, |E| = n,and no loop, then 2n ≤ m2 − m b) Since (v1, v2) ≠ (v2, v1)for digraph without loops, we have n ≤ m2 − m -101- 11.2 Complements and Graph Isomor- phism -study the structure of graphs Def: G = (V, E)(directed or undirected) U ⊆ V, U is not empty The subgraph of G induced by U is the sub- graph with vertices in U and all edges (from G)ofthe following form (a) (x, y), x, y ∈ U (G directed), or (b) {x, y}, x, y ∈ U (G undirected) This subgraph is denoted by <U> Asubgraph G′ of a graph G = (V, E)iscalled an induced subgraph if there exists U⊆V such that G′=<U> Note: An induced subgraph is a subgraph But a subgraph is not necessary to be an induced subgraph. Why? -102- Def: G = (V, E)(directed or undirected) G − v = (V −{v}, E′), E′ contains all edges of G except those that are incident with v G − e = (V, E −{e}) Def: |V| = n Kn:(the complete graph on V)isaloop-free undirected graph where V- a, b ∈ V, a ≠ b, there is an edge {a, b} n2 − n (Hence, number of edges of K = ) n 2 Examples: n = 4, n = 5 Def: G = (V, E): loop-free, undirected, |V| = n Kn = (V, E′): complete graph on V The Complement of G is defined as follows: G = (V, E′−E) -103- . K4 G G Def: G1 = (V1, E1), G2 = (V2, E2): undirected f : V1 → V2 is a graph isomorphism if (a) f is one-to-one and onto (b) {a, b} ∈ E1 iff{f (a), f (b)} ∈ E2 In this case, G1 and G2 are called isomorphic graphs (i.e., G1 and G2 have the same struc- ture). Example: a isomorphic q f w b r . e v x s j g h z i t d c u y a → q, b → v, c → u, d → y, e → r f → w, g → x, h → t, i → z, j → s -104- Notes: An isomorphism preserves adjacencies, hence, structures such as "paths" and "cycles". Howtodetermine if twographs are isomorphic? -Ifacycle in G1 does not have a counter part in G2 then G1 can not be isomorphic to G2. -Ifapath in G1 does not have a counter part in G2 then G1 can not be isomorphic to G2. -Degrees of adjancyofcorresponding vertices in isomorphic graphs must be the same. Example: The following twographs are not isomorphic. Why? . -105- The number of vertices with degree of adjancy2 is 2 in G1 butthe that number in G2 is 3, or The number of vertices with degree of adjancy4 is 2 in G1 butthe that number in G2 is 3, or Each vertexof G2 can be the start point of a trail which includes every edge of the graph. But in G1, f and b are the only vertices with such a property. Example: If every induced subgraph of G = (V, E), |V| ≥ 2isconnected then G is isomorphic to Kn where n = |V|. (Prove that |E| = (n2 − n)/2. If |E|<(n2 − n)/2 then it is possible to find an induced subgraph of G with twoelements which is not connected.) -106- Example: Find all (loop-free) non-isomorphic undi- rected graphs with four vertices. Howmany of them are connected? 0edge 1edge 2edges 3edges 4edges 5edges 6edges.