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Graphs CSCI 2720 Spring 2005

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

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

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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.

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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.

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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.

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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)}.

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Definition: walk, length, path, cycle A walk in a graph (digraph) G is a sequence of vertices v 0, v 1, … v n such that, for all 0 <= i< n, (v i, v i+1 ) is an edge (arc) in G. The length of the walk v 0, v 1, … v n 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 v 0 = v n and no other vertex is repeated; sometimes, if it is understood, we omit v n from the sequence.

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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

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

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

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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).

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

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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 ).

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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)

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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)

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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.

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

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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.

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Adjacency matrices for G1,G2

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

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Matrix vs. list representation Matrix n vertices and m edges requires O( n 2 ) 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

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