# 1.7. Isomorphic Graphs

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1 1.7. Isomorphic Graphs Example: Consider the following graphs, are they the isomorphic, i.e. the same? No. The left-hand graph has 5 edges; the right hand graph has 6 edges. WUCT121 Graphs 25

2 Firstly, label the graphs. It looks true, so check all the things we know: v 5 v 4 v v' 1 1 v 3 v 2 v' 4 v' 2 G G v ' 3 v ' 5 Number of vertices: both 5. Number of edges: both 5. Degrees of corresponding vertices: all degree 2. Connectedness: Each is fully connected. Number of connected components: Both 1. Pairs of connected vertices: All correspond. Number of loops: 0. Number of parallel edges: 0. Everything is equal and so the graphs are isomorphic. WUCT121 Graphs 26

3 More formally: G = { V, E} where V = v, v, v, v, } and E = {( v = { e 1 1, v, e 2 2 ), ( v, e G = { V, E } 3 2, e 4, v 3, e 5 { v5 ),( v } 3, v 4 ),( v 4, v 5 ),( v where V = { v, v, v, v v } E = {( v, v = { e, e, e, e, e } ), ( v , 5 5, v and, v ),( v, v 3 4 ),( v 4, v 5 1 )} ),( v5, v1 )}. Construct 2 functions: f : V V and g : E E f : V V g : E E V V E E v 1 v 1 e 1 e 1 v 2 v 2 e 2 e 2 v 3 v 3 e 3 e 3 v 4 v 4 e 4 e 4 v 5 v 5 e 5 e 5 WUCT121 Graphs 27

4 Definition Let G = { V, E} and G = { V, E } be graphs. G and G are said to be isomorphic if there exist a pair of functions f : V V and g : E E such that f associates each element in V with exactly one element in V and vice versa; g associates each element in E with exactly one element in E and vice versa, and for each v V, and each e E, if v is an endpoint of the edge e, then f (v) is an endpoint of the edge g (e). Notes: To prove two graphs are isomorphic you must give a formula (picture) for the functions f and g. If two graphs are isomorphic, they must have: - the same number of vertices - the same number of edges - the same degrees for corresponding vertices - the same number of connected components - the same number of loops WUCT121 Graphs 28

5 - the same number of parallel edges. Further, - both graphs are connected or both graphs are not connected, and - pairs of connected vertices must have the corresponding pair of vertices connected. In general, it is easier to prove two graphs are not isomorphic by proving that one of the above properties fails. WUCT121 Graphs 29

6 Exercises: Show that the following graphs are isomorphic. WUCT121 Graphs 30

7 Draw all possible graphs having 2 edges and 2 vertices; that is, draw all non-isomorphic graphs having 2 edges and 2 vertices. Since isomorphic graphs are essentially the same, we can use this idea to classify graphs. WUCT121 Graphs 31

8 1.8. Regular, Complete and Complete Bipartite Definition: Regular. A simple graph G = { V, E} is said to be regular of degree k, or simply k-regular if for each v V, δ ( v) = k. That is, if a graph is k-regular, every vertex has degree k. Exercises: Draw all 0-regular graphs with 1 vertex; 2 vertices; 3 vertices. 1 vertex: 2 vertices: 3 vertices: Draw all 1-regular graphs with 1 vertex; 2 vertices; 3 vertices; 4 vertices. 1 vertex and 3 vertices not possible WUCT121 Graphs 32

9 Draw all 2-regular graphs with 2 vertices; 3 vertices; 4 vertices Definition: Complete. A simple graph G = { V, E} is said to be complete if each vertex of G is connected to every other vertex of G. The complete graph with n vertices is denoted K n. Notes: A complete graph is connected isomorphic n, two complete graphs having n vertices are For complete graphs, once the number of vertices is known, the number of edges and the endpoints of each edge are also known WUCT121 Graphs 33

10 Example: Draw K 5 v 5 v 1 v 2 v 4 v 3 Exercises: Draw K 1, K 2, K 3 and K 4. What is the degree of each vertex in each of the graphs drawn? WUCT121 Graphs 34

11 Complete the following sentences: o A complete graph, K n, is an -regular graph. o A complete graph, K n, has [Use Euler s First Law: ( v) = 2n( E) v V δ.] edges. WUCT121 Graphs 35

12 Definition: Bipartite. A simple graph G = { V, E} is said to be bipartite if there exists sets Notes: U V and W V, such that; 1. U W = V and U W = 2. every edge of G connects a vertex in U with a vertex in W. A bipartite graph G = { V, E} is one whose vertices can be separated into two disjoint sets, where every edge joins a vertex in one set to a vertex in the other u U, δ ( u) n( W ) and w W, δ ( w) n( U ) [ n (W ) is the number of elements in W] WUCT121 Graphs 36

13 Example: Consider the following graph G = { V, E}, where V = { u1, u2, w1, w2, w3}, is it bipartite? Yes u 1 u2 e 1 e 2 e 3 e 4 List the sets U and W. U = u 1, u } ; W = w, w, } Exercises: { 2 { 1 2 w3 Consider the following graphs, are they bipartite? If so write down the sets U and W. o w 1 w2 w3 u 1 u2 e 1 e2 w 1 w2 WUCT121 Graphs 37

14 o u 1 w 2 w 1 u 2 Draw two bipartite graphs with 5 vertices. o Where U has 1 element and W has 4 elements. o Where U has 2 elements and W has 3 elements. WUCT121 Graphs 38

15 Definition: Complete Bipartite. A simple graph G = { V, E}, is said to be complete bipartite if; Notes: 1. G is bipartite and 2. every vertex in U is connected to every vertex in W. A complete bipartite graph is one whose vertices can be separated into two disjoint sets where every vertex in one set is connected to every vertex in the other but no vertices within either set are connected. The symbol K n, m is used to denote the complete bipartite graph having n vertices in one set and m in the other. [Note that the comma is necessary!] u U, δ ( u) = n( W ) and w W, δ ( w) = n( U ) n ( E) = n m WUCT121 Graphs 39

16 Examples: Draw K 2, 2 Draw K 2, 3 K 2, 2 K 2,3 Exercises: Draw K 1, 3 Draw K 2, 4 WUCT121 Graphs 40

17 Which of these graphs are bipartite. WUCT121 Graphs 41

18 1.9. Eulerian Graphs. Problem 1: If you are walking in the park, can you cross each bridge exactly once? Does it depend on where you start? Could you do it with less bridges? Could you do it with more bridges? Problem 2: The Koenigsberg Bridge Problem. WUCT121 Graphs 42

19 Can you stroll around, crossing each bridge exactly once and return to your starting point? No one can do it. How do you prove it can't be done? If you're Leonhard Euler, you make some definitions, make the problem precise and prove a theorem! Redraw the problem as a graph: Definition: Circuit. Let G = { V, E} be a graph and let v V. A circuit in G is a path from v to v in which no edge is repeated. Note: A circuit is a closed path and in many books is called a cycle. WUCT121 Graphs 43

20 Exercises: Write down the circuits in the following graph, starting at the given vertices. o v 2 : o v 1 : o v 3 : o v 7 : Definition: Eulerian Path Let G = { V, E} be a graph. A path in G is an Eulerian path if every edge of G is included once and only once in the path. WUCT121 Graphs 44

21 Definition: Eulerian Circuit Let G = { V, E} be a graph. A circuit in G is an Eulerian circuit if every edge of G is included exactly once in the circuit Definition: Eulerian Graph Let G = { V, E} be a graph. G is an Eulerian graph if G has an Eulerian circuit. Exercises: Which of these graphs are Eulerian? K 3 v 2 v 1 v 3 v 1 v 3 K 4 v v 2 4 v 5 v 1 v 2 K 5 v 4 v K 3 2, 3 WUCT121 Graphs 45

22 Can you find an Eulerian path in the following graph that is not an Eulerian circuit? v 1 e 1 v 2 e 6 e 4 e 2 v 3 v 5 e 5 v 4 e 3 Is the graph Eulerian? In the Koenisberg Bridge Problem, we wanted to start and end at the same vertex. That is, we need to prove that the graph K is not Eulerian. How? Theorem: Euler's Theorem. Let G be a connected graph. G is an Eulerian graph if and only if the degree of each vertex is even. WUCT121 Graphs 46

23 For the Koenisberg Bridge Problem: δ ( C) = 3, so the graph is not Eulerian. Discussion: Is the connectedness condition in Euler's Theorem necessary? Consider the graph Each vertex has even degree but there is no Eulerian circuit. WUCT121 Graphs 47

24 In the Koenisberg Bridge setup, is it possible to cross each bridge exactly once without necessarily finishing where you start? That is, is there an Eulerian path in K? Theorem: Existence of Eulerian Path. A connected graph has an Eulerian path, which is not a circuit, if it has exactly two vertices of odd degree. Example: δ ( A) = 3; δ ( B) = 5; δ ( C) = 3; δ ( D) = 3 There are more than 2 vertices of odd degree, therefore, K does not have an Eulerian path. WUCT121 Graphs 48

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