Graphs

1. Graphs CSCI 2720 Spring 2005

2. Graph • Why study graphs? • important for many real-world applications • compilers • Communication networks • Reaction networks • & more

3. The Graph ADT • a set of nodes (vertices or points) • connection relations (edges or arcs) between those nodes • Definitions follow ….

4. Definition : graph • A graph G=(V,E) is • a finite nonempty set V of objects called vertices (the singular is vertex) • together with a (possibly empty) set E of unordered pairs of distinct vertices of G called edges. • Some authors call a graph by the longer term ``undirected graph'' and simply use the following definition of a directed graph as a graph. However when using Definition 1 of a graph, it is standard practice to abbreviate the phrase ``directed graph'' (as done below in Definition 2) with the word digraph.

5. Definition: digraph • A digraph G=(V,E) is • a finite nonempty set V of vertices • together with a (possibly empty) set E of ordered pairs of vertices of G called arcs. • An arc that begins and ends at a same vertex u is called a loop. We usually (but not always) disallow loops in our digraphs. • By being defined as a set, E does not contain duplicate (or multiple) edges/arcs between the same two vertices. • For a given graph (or digraph) G we also denote the set of vertices by V(G) and the set of edges (or arcs) by E(G) to lessen any ambiguity.

6. Definition: order, size • The order of a graph (digraph) G=(V,E) is |V|, sometimes denoted by |G| , and the size of this graph is |E| . • Sometimes we view a graph as a digraph where every unordered edge (u,v) is replaced by two directed arcs (u,v) and (v,u) . In this case, the size of a graph is half the size of the corresponding digraph.

7. Example • G1 is a graph of order 5 • G2 is a digraph of order 5 • The size of G1 is 6 where E(G1) = • {(0, 1), (0, 2), (1, 2), (2, 3), (2, 4), (3, 4)} • The size of the digraph G2 is 7 where E(G2) = • {(0, 2), (1, 0), (1, 2), (1, 3), (3, 1), (3, 4), (4, 2)}.

8. Definition: walk, length, path, cycle • A walk in a graph (digraph) G is • a sequence of vertices v0, v1, … vn such that, for all 0 <= i< n , (vi, vi+1) is an edge (arc) in G . • The length of the walk v0, v1, … vn is • the number n (i.e., number of edges/arcs). • A path is • a walk in which no vertex is repeated. • A cycle is • a walk (of length at least three for graphs) in which v0 =vn and no other vertex is repeated; sometimes, if it is understood, we omit vn from the sequence.

9. Example walks • Walks in G1: • 0,1,2, 3, 4 • 0,1,2,0 • 0,1,2 • 0,1,0 • Walks in G2: • 3,1,2 • 1,3,1 • 3,1,3,1,0

10. Example paths • Paths in G1: • 0,1,2, 3, 4 • 0,1,2 • Paths in G2: • 3,1,2

11. Example cycles • Cycles in G1: • 0,1,2,0 • 0,1,2 (understood) • Cycles in G2: • 1,3,1

12. Definition: connected, strongly connected • A graph G is connected if • there is a path between all pairs of vertices u and v of V(G) . • A digraph G is strongly connected if • there is a path from vertex u to vertex v for all pairs u and v in V(G).

13. Connected? • G1 is connected • G2 is not strongly connected. • No arcs leaving vertex 2

14. Definition: degree • In a graph, the degree of a vertex v , denoted by deg(v), is • the number of edges incident to v . • in-degree == out-degree • For digraphs, the out-degree of a vertex v is • the number of arcs {(v,z) € E| z € V} incident from v (leaving v ) and the in-degree of vertex v is the number of arcs {(z,v) € E| z € V} incident to v (entering v ).

15. Degree, degree sequence • G1: • deg(0) = 2 • deg(1) = 2 • deg(2) = 4 • deg(3) = 2 • Deg(4) = 2 • Degree sequence = (2,2,4,2,2)

16. Degree, degree sequence • G2: • In-degree sequence = (1,1,3,1,1) • Out-degree sequence = (1,3,0,2,1) • Degree of vertex of a digraph sometimes written as sum of in-degree and out-degree: • (2,4,3,3,2)

17. Definition: diameter • The diameter of a connected graph or strongly connected digraph G=(V,E) is • the least integer D such that for all vertices u and v in G we have d(u,v) <=D, where d(u,v) denotes the distance from u to v in G, that is, the length of a shortest path between u and v.

18. Diameter • G1: • min(d(u,v) )= 2 • Diameter = 2 • G2: not strongly connected, diameter not defined

19. Computer representations • adjacency matrices • For a graph G of order n , an adjacency matrix representation is a boolean matrix (often encoded with 0's and 1's) of dimension n such that entry (i,j) is true if and only if edge/arc (I,j) is in E(G). • adjacency lists • For a graph G of order n , an adjacency lists representation is n lists such that the i-th list contains a sequence (often sorted) of out-neighbours of vertex i of G .

20. Adjacency matrices for G1,G2

21. Adjacency lists for G1,G2 For digraphs, stores only the out-edges

22. Matrix vs. list representation • Matrix • n vertices and m edges requires O( n2 ) storage • check if edge/arc (i,j) is in graph – O(1) • List • n vertices and m edges, requires O(m) storage • Preferable for sparse graphs • tcheck if edge/arc (i,j) is in graph - O(n) time • Note: other specialized representations exist