Algorithmic Bioinformatics III Graphs, Networks, and Systems SS2008 Ralf Zimmer

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1 Algorithmic Bioinformatics III Graphs, Networks, and Systems SS2008 Ralf Zimmer Graph Theory Introduction Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 1

2 Graphs Graph G = (V, E) V vertices E edges F: E -> V & V (undirected) Incidence matrix Digraph G = (V, A) A arcs F: E -> V x V (directed) Graph trivial V = 0 Graph isolated E = 0 G1 ~ G2 isomorph There is a bijection f: V1 -> V2 conserving the incidences G planar nodes embedded in plane without crossing of edges Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 2

3 Graph Examples A A B C B C K 3 = (V,E) V = {A,B,C} E = {{A,B}, {A,C}, {B,C}} Directed graph (Digraph) E = {(A,B), (A,C), (C,B)} Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 3

4 Graphs: Basic Definitions Given a graph G = (V,E) v Def.: Incidence Vertex v: Incidence(v) = { e = (x,y) in E x = v or y = v } Def.: Vertex Adjacence Vertex v: Adjacence(v) = { w in V (v,w) or (w,v) in E } Def.: Edge Adjacence Edge e: Adjacence(e) = { e in E e and e share a vertex } = { e = (x,y) in E e = (v,w), x = v or x =w or y = v or y = w } v e w Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 4

5 Graphs: Basic Definitions Def.: start/end incidence with v (Digraph) Vertex v: Start Incidence(v) = {e = (v,y) in E} End Incidence(v) = {e = (y,v) in E} v Def.: start/end incidence with A (Digraph) Set of vertices A subset of V: Start-Incidence(A) = {e = (x,y) in E x in A, y not in A} End-Incidence(A) = {e = (y,x) in E x in A, y not in A} v ω+(a) := set of edges start incident with A ω (A) := set of edges end incident with A ω(a) := set of edges incident with A e A Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 5

6 Graphs: Basic Definitions Representations for graphs G = (V, E), V = n, E = m Adjacency Matrix: n x n matrix V V Incidence Matrix: n x m matrix E V Edge list [e e in E] Adjacency list [v, Adjacence(v) v in V] Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 6

7 Graphs: Basic Definitions Def.: C Induced subgraph C is a subset of nodes and all incident edges Def.: Subgraph Subset of nodes and edges Def.: Spanning subgraph All vertices and a subset of edges (not necessarily connected) Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 7

8 Graphs: Basic Definitions Def.: Digraph D symmetric / antisymmetric D symmetric: (v,w) in E => (w,v) in E D antisymmetric: (v,w) in E => (w,v) not in E Def.: A graph is Complete Directed: (v,w) not in E => (w,v) in E Un-directed: Forall v,w in V: (v,w) in E Def.: (Un-)Directed walk w := sequence of edges: e 1 e 2 e n v 0 v 1 v 2 v n-1 v n w = (e 1 =(v 0,v 1 ),, e i =(v i-1,v i ),, e k =(v k-1,v k ) ) e i = (x,y) in E or e i :=(y,x) in E Path: edges of walk disjunct Simple path: vertices on path disjunct Def.: G connected: forall v,w in V there is a path v -> w Def.: G strongly connected: forall v,w in V there is a directed path v -> w Def.: Connected component C of G = connected induced subgraph of G Def.: A graph G is connected G has only one connected component Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 8

9 Graphs: Cycles and Co-cycles Def.: Cycle := closed path (e1,, ek) = μ e 1 = (v,y), e k = (x,v) e i = (x,y) or e i = (y,x) in E Def.: Elementary cycle := closed simple path (e1,, ek) Vertices on path are disjunct μ+ subset μ: (edges with same direction as e1 in cycle μ) μ = μ μ+ Def.: Degree(v) = deg(v) = δ(v) := number of edges incident with v incidence(v) v Degree(v) = 4 Def.: μ circuit μ empty Def.: Incidence vector of cycle: μ = (μ 1,, μ m ), m= E μ i = 0 (e i not in cycle μ), μ i = 1(e i in μ+), μ i = -1 (e i in μ ), Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 9

10 Graph Isomorphism G1 = (V1,E1), G2 = (V2,E2) G1 isomorph G2 exists function f, which maintains the incidence structure V1 = V2 E1 = E2 Bijection s.t. Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 10

11 Graph Isomorphism Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 11

12 Graph Isomorphism Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 12

13 Graph Isomorphism Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 13

14 Graph Isomorphism Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 14

15 Subgraph Isomorphism Def: (Subgraph isomorphism) [SGI] Instance: G =(V,E), H=(V,E ) Problem: Does G contain a subgraph isomorphic to H i.e. there are subsets V of V and E of E such that V = V and E = E and a bijection f: V -> V, which observes the incidences: {u,v} in E {f(u),f(v)} in E Theorem: [SGI] is NP-complete. Theorem: [SGI] is NP-complete for directed graphs. Theorem: [SGI] is NP-complete even if G is acyclic and H is a directed tree. Whether Graph Isomorphism [GI] is NP-complete for un-directed or for directed graphs are open problems. Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 15

16 Subgraph Isomorphism Def: (Subgraph isomorphism) [SGI] Instance: G =(V,E), H=(V,E ) Problem: Does G contain a subgraph [GI] is isomorphic polynomial to for H special graph classes: i.e. there are subsets V of V and E of E such that Trees V = V and E = E and a bijection f: V -> V, which Planar graphs observes the incidences: {u,v} in E {f(u),f(v)} in E Interval graphs Permutation graphs Theorem: [SGI] is NP-complete. Partial k-trees Bounded-parameter graphs Theorem: [SGI] is NP-complete for directed graphs. Theorem: [SGI] is NP-complete even if G is acyclic and H is a directed tree. Whether Graph Isomorphism [GI] is NP-complete for un-directed or for directed graphs are open problems. Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 16

17 Planar Graphs Graphs cannot always be embedded into R 2 without crossings But always in R 3! Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 17

18 Planar Graphs Graphs cannot always be embedded into R 2 without crossings But always in R 3! K 5 is not planar K 33 not planar Theorem (Kuratowski) A graph G is not planar G contains K 5 or K 33 as a homeomorphic subgraph Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 18

19 Def: G and G are homeomorphic if there is a subdivision of G and a subdivision of G which are isomorphic Def: A graph H is a subdivision of a graph G the edges set E(H) of H can be obtained from the edges of G by subdivision: Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 19

20 Euler Formula Theorem: For a connected, planar graph G = (V, E) with V = n, E =m, f number of faces, then n + f = m + 2 (1) (3) (2) (2) Proof: m=0: n=1 and f=1 Ok. m-1->m: (1) m loop: n =n, f =f+1 and m =m+1 (2) splitting a face: n =n, f =f+1 and m =m+1 (3) new node: n =n+1, f =f and m =m+1 Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 20

21 Def.: (Bridge) An edge of a connected graph G is called a bridge G= (V,E-{e}) is not connected Def.: (Girth) The girth of a connected graph is the length of its shortest cycle Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 21

22 Planarity - girth Theorem: G a connected, planar graph on n vertices. If G has girth at least g, then G can have at most g(n-2)/(g-2) edges. (If G is acyclic => G has exactly n-1 edges) Proof: n >= 3 Induction on n I.A.: n=3 => m <= 3 = 3(3-2)/(3-2) Ok. I.S.: (i) G contains a bridge e => remove bridge e, s.t. G is divided in two connected subgraphs G1 and G2. Then G1 or G2 (or both) contain a cycle (as e is a bridge). Both: G2 acyclic (m2 = n2-1) Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 22

23 Planarity - girth Theorem: G a connected, planar graph on n vertices. If G has girth at least g, then G can have at most g(n-2)/(g-2) edges. (If G is acyclic => G has exactly n-1 edges) Proof: (contd.) Induction on n (contd.) I.S.: (ii) G does not contain a bridge e => each edge e of G contains to exactly two faces: (f_i = # of faces with a border of i edges) (as each cycle contains at least g edges) (Euler Theorem) i.e. Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 23

24 Planarity - girth Theorem: G a connected, planar graph on n vertices. If G has girth at least g, then G can have at most g(n-2)/(g-2) edges. (If G is acyclic => G has exactly n-1 edges) Since every planar graph is acyclic or has girth at least 3, we have: Lemma: A connected planar graph with n vertices (n >=3) has at most 3n 6 edges Proof Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 24

25 K 5 Lemma: For a planar graph, we have: 3n >= m+6, i.e. E <= 3 V - 6 Theorem: K 5 is not planar Proof: Assumption: K 5 is planar => Euler formula holds: n+f=m+2 m=10, n=5 3f <= 2m, at least 3 edges are necessary per face in a planar graph without multiple edges n+f = m+2 3n+3f = 3m+6 3n+2m >= 3m+6 3n >= m+6 3*5 >= 10+6 contradiction Def: The girth g(g) of a graph G is the length of the shortest cycle of G g(k 5 ) = 3 Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 25

26 K 33 Theorem: K 33 is not planar Proof: Assumption: K 33 is planar => Euler theorem holds m=9, n=6 4f <= 2m, only cycles of even length >= 4 (bipartite!) n+f = m+2 f = m+2-n = = 5 4f = 4*5 = 20 <= 2*m = 18 Contradiction Def: The girth g(g) of a graph G is the length of the shortest cycle of G g(k 3,3 ) = 4 Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 26

27 Kuratowski s Theorem Theorem: A graph G is nonplanar if and only if there is a subgraph of G which is homeomorphic to either K 33 or K 5 Proof: <= If G contains such a subgraph it is not planar => quite lengthy, see [Even, 1979, p ] G1 and G2 isomorph: G1 planar G2 planar G1 and G2 homeomorph if both can be obtained from the same graph by the insertion of new vertices of degree 2 into edges, an edge is replaced by a path whose intermediate vertices are all new. Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 27

28 Kuratowski s Theorem Theorem: A graph G is nonplanar if and only if there is a subgraph of G which is homeomorphic to either K 33 or K 5 Kuratowski s theorem provides a sufficient and necessary condition for a graph to be planar. => efficient algorithm for planarity testing? Algorithm: Try all 5 vertex subsets and check whether it induces a complete subgraph (homeomorphic: contains 10 vertex disjoint paths connecting all pairs) Try all pairs of 3 vertices whether it induces a complete subgraph Problems: There are V V V 3 and such subsets (O( V 6 ). 5 1/ Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 28

29 Kuratowski s Theorem Theorem: A graph G is nonplanar if and only if there is a subgraph of G which is homeomorphic to either K 33 or K 5 Kuratowski s theorem provides a sufficient and necessary condition for a graph to be planar. => efficient algorithm for planarity testing? Algorithm: Try all 5 vertex subsets and check whether it induces a complete subgraph (homeomorphic: contains 10 vertex disjoint paths connecting all pairs) Try all pairs of 3 vertices whether it induces a complete subgraph Problems: V V V 3 There are and such subsets (O( V 6 ). 5 1/ 2 Check for disjoint paths is difficult (exponential) There are linear (O( E )) algorithms for testing planarity! 3 [Hopcroft&Tarjan: Efficient Planarity Testing, JACM, Vol.21, No.4, 1974, ] 3 NP-hard Garey&Johnson [ND40] Ralf Zimmer, LMU Institut für Informatik, Lehrstuhl für Praktische Informatik und Bioinformatik, SS 2008: Algorithmic Bioinformatics III: Graphs and Networks 29