Discrete Mathematics Graphs H. Turgut Uyar Ay seg ul Gen cata - PowerPoint PPT Presentation

Walk Example ( c , b ) ( b , a ) ( a , d ) − − − → b − − − → a − − − → d c ( d , e ) ( e , f ) ( f , a ) − − − → e − − − → f − − − → a ( a , b ) − − − → b c → b → a → d → e → f → a → b length: 7

Trail trail: edges not repeated circuit: closed trail spanning trail: covers all edges

Trail Example ( c , b ) ( b , a ) ( a , e ) − − − → b − − − → a − − − → e c ( e , d ) ( d , a ) ( a , f ) − − − → d − − − → a − − − → f c → a → e → d → a → f

Path path: nodes not repeated cycle: closed path spanning path: visits all nodes

Path Example ( c , b ) ( b , a ) ( a , d ) − − − → b − − − → a − − − → d c ( d , e ) ( e , f ) − − − → e − − − → f c → b → a → d → e → f

Connected Graphs connected: a path between every pair of nodes a disconnected graph can be divided into connected components

Connected Components Example disconnected: no path between a and c connected components: a , d , e b , c f

Distance distance between v i and v j : length of shortest path between v i and v j diameter of graph: largest distance in graph

Distance Example distance between a and e : 2 diameter: 3

Cut-Points G − v : delete v and all its incident edges from G v is a cut-point for G : G is connected but G − v is not

Cut-Point Example G G − d

Directed Walks ignoring directions on arcs: semi-walk , semi-trail , semi-path if between every pair of nodes there is: a semi-path: weakly connected a path from one to the other: unilaterally connected a path: strongly connected

Directed Graph Examples weakly unilaterally strongly

Topics 1 Graphs Introduction Walks Traversable Graphs Planar Graphs 2 Graph Problems Connectivity Graph Coloring Shortest Path TSP Searching Graphs

Bridges of K¨ onigsberg cross each bridge exactly once and return to the starting point

Graphs

Traversable Graphs G is traversable: G contains a spanning trail a node with an odd degree must be either the starting node or the ending node of the trail all nodes except the starting node and the ending node must have even degrees

Bridges of K¨ onigsberg all nodes have odd degrees: not traversable

Traversable Graph Example a , b , c : even d , e : odd start from d , end at e : d → b → a → c → e → d → c → b → e

Euler Graphs Euler graph: contains closed spanning trail G is an Euler graph ⇔ all nodes in G have even degrees

Euler Graph Examples not Euler Euler

Hamilton Graphs Hamilton graph: contains a closed spanning path

Hamilton Graph Examples Hamilton not Hamilton

Topics 1 Graphs Introduction Walks Traversable Graphs Planar Graphs 2 Graph Problems Connectivity Graph Coloring Shortest Path TSP Searching Graphs

Planar Graphs Definition G is planar: G can be drawn on a plane without intersecting its edges a map of G : a planar drawing of G

Planar Graph Example

Regions map divides plane into regions degree of region: length of closed trail that surrounds region Theorem d r i : degree of region r i � i d r i | E | = 2

Regions map divides plane into regions degree of region: length of closed trail that surrounds region Theorem d r i : degree of region r i � i d r i | E | = 2

Region Example d r 1 = 3 = 3 d r 2 d r 3 = 5 = 4 d r 4 d r 5 = 3 = 18 | E | = 9

Euler’s Formula Theorem (Euler’s Formula) G = ( V , E ) : planar, connected graph R: set of regions in a map of G | V | − | E | + | R | = 2

Euler’s Formula Example | V | = 6, | E | = 9, | R | = 5

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 | E | ≤ 3 | V | − 6 Proof. sum of region degrees: 2 | E | degree of a region ≥ 3 ⇒ 2 | E | ≥ 3 | R | ⇒ | R | ≤ 2 3 | E | | V | − | E | + | R | = 2 ⇒ | V | − | E | + 2 3 | E | ≥ 2 ⇒ | V | − 1 3 | E | ≥ 2 ⇒ 3 | V | − | E | ≥ 6 ⇒ | E | ≤ 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 : ∃ v ∈ V [ d v ≤ 5] Proof. assume: ∀ v ∈ V [ d v ≥ 6] ⇒ 2 | E | ≥ 6 | V | ⇒ | E | ≥ 3 | V | ⇒ | E | > 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 : ∃ v ∈ V [ d v ≤ 5] Proof. assume: ∀ v ∈ V [ d v ≥ 6] ⇒ 2 | E | ≥ 6 | V | ⇒ | E | ≥ 3 | V | ⇒ | E | > 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 : ∃ v ∈ V [ d v ≤ 5] Proof. assume: ∀ v ∈ V [ d v ≥ 6] ⇒ 2 | E | ≥ 6 | V | ⇒ | E | ≥ 3 | V | ⇒ | E | > 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 : ∃ v ∈ V [ d v ≤ 5] Proof. assume: ∀ v ∈ V [ d v ≥ 6] ⇒ 2 | E | ≥ 6 | V | ⇒ | E | ≥ 3 | V | ⇒ | E | > 3 | V | − 6

Planar Graph Theorems Theorem G = ( V , E ) : connected, planar graph where | V | ≥ 3 : ∃ v ∈ V [ d v ≤ 5] Proof. assume: ∀ v ∈ V [ d v ≥ 6] ⇒ 2 | E | ≥ 6 | V | ⇒ | E | ≥ 3 | V | ⇒ | E | > 3 | V | − 6

Nonplanar Graphs Theorem K 5 is not planar. Proof. | V | = 5 3 | V | − 6 = 3 · 5 − 6 = 9 | E | ≤ 9 should hold but | E | = 10

Nonplanar Graphs Theorem K 5 is not planar. Proof. | V | = 5 3 | V | − 6 = 3 · 5 − 6 = 9 | E | ≤ 9 should hold but | E | = 10

Nonplanar Graphs Theorem K 5 is not planar. Proof. | V | = 5 3 | V | − 6 = 3 · 5 − 6 = 9 | E | ≤ 9 should hold but | E | = 10

Nonplanar Graphs Theorem K 5 is not planar. Proof. | V | = 5 3 | V | − 6 = 3 · 5 − 6 = 9 | E | ≤ 9 should hold but | E | = 10

Nonplanar Graphs Theorem K 5 is not planar. Proof. | V | = 5 3 | V | − 6 = 3 · 5 − 6 = 9 | E | ≤ 9 should hold but | E | = 10

Nonplanar Graphs Theorem Proof. K 3 , 3 is not planar. | V | = 6 , | E | = 9 if planar then | R | = 5 degree of a region ≥ 4 ⇒ � r ∈ R d r ≥ 20 | E | ≥ 10 should hold but | E | = 9

Nonplanar Graphs Theorem Proof. K 3 , 3 is not planar. | V | = 6 , | E | = 9 if planar then | R | = 5 degree of a region ≥ 4 ⇒ � r ∈ R d r ≥ 20 | E | ≥ 10 should hold but | E | = 9

Nonplanar Graphs Theorem Proof. K 3 , 3 is not planar. | V | = 6 , | E | = 9 if planar then | R | = 5 degree of a region ≥ 4 ⇒ � r ∈ R d r ≥ 20 | E | ≥ 10 should hold but | E | = 9

Nonplanar Graphs Theorem Proof. K 3 , 3 is not planar. | V | = 6 , | E | = 9 if planar then | R | = 5 degree of a region ≥ 4 ⇒ � r ∈ R d r ≥ 20 | E | ≥ 10 should hold but | E | = 9