2018 - Page 13 of 41 - Shawn Zhong

10.4 Connectivity

Paths • Informal Definition ○ A path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph. ○ As the path travels along its edges, it visits the vertices along this path, that is, the endpoints of these. • Applications ○ Numerous problems can be modeled with paths formed by traveling along edges of graphs such as ○ determining whether a message can be sent between two computers. ○ efficiently planning routes for mail delivery. • Definition ○ Let n be a nonnegative integer and G an undirected graph ○ A path of length n from u to v in G is a sequence of n edges e_1,…,e_n of G for which § there exists a sequence x_0=u,x_1,…,x_(n−1),x_n=v of vertices such that § e_i has, for i=1,…,n, the endpoints x_(i−1) and x_i. ○ When the graph is simple, we denote this path by its vertex sequence § x_0,x_1,…,x_n (since listing the vertices uniquely determines the path). ○ The path is a circuit if it begins and ends at the same vertex and has length greater than zero. ○ The path or circuit is said to pass through the vertices x_1,x_2,…,x_(n−1) and traverse the edges e_1,…,e_n. ○ A path or circuit is simple if it does not contain the same edge more than once. • Example ○ In the simple graph here: ○ a, d, c, f, e is a simple path of length 4. ○ d, e, c, a is not a path because e is not connected to c. ○ b, c, f, e, b is a circuit of length 4. ○ a, b, e, d, a, b is a path of length 5, but it is not a simple path. Degrees of Separation • Paths in Acquaintanceship Graphs ○ In an acquaintanceship graph there is a path between two people if there is a chain of people linking these people, where two people adjacent in the chain know one another. ○ In this graph there is a chain of six people linking Kamini and Ching. • Note: ○ Some have speculated that almost every pair of people in the world are linked by a small chain of no more than six, or maybe even, five people. ○ The play Six Degrees of Separation by John Guare is based on this notion. Connectedness in Undirected Graphs • Definition ○ An undirected graph is called connected if there is a path between every pair of vertices. ○ An undirected graph that is not connected is called disconnected. ○ We say that we disconnect a graph when we remove vertices or edges, or both, to produce a disconnected subgraph. • Example ○ G_1 is connected because there is a path between any pair of its vertices, as can be easily seen. ○ However G_2 is not connected because there is no path between vertices a and f, for example. • Theorem ○ Every pair of distinct vertices in a connect graph is connected by a simple path • Proof ○ Let u and v be two distinct vertices of the connected undirected graph G=(V,E) ○ Because G is connected, there is at least one path between u and v ○ Let x_0,x_1,…,x_n, where x_0=u and x_n=v, be the vertex sequence of a path of least length. ○ This path of least length is simple. ○ To see this, suppose is not simple, then x_i=x_j for some i and j with 0≤i j ○ This means that there is a path from u to v of shorter length with vertex sequence § x_0,x_1,…,x_(i−1),x_j,…,x_n ○ Obtained by deleting the edges corresponding to the vertex sequence x_i,…,x_(j−1) Connected Components Definition A connected component of a graph G is a connected subgraph of G that is not a proper subgraph of another connected subgraph of G A graph G that is not connected has two or more connected components that are disjoint and have G as their union. Example The graph H is the union of three disjoint subgraphs H_1, H_2, and H_3, none of which are proper subgraphs of a larger connected subgraph of G These three subgraphs are the connected components of H. Connectedness in Directed Graphs • Definition ○ A directed graph is strongly connected if there is a path from a to b and a path from b to a whenever a and b are vertices in the graph. • Definition ○ A directed graph is weakly connected if there is a path between every two vertices in the underlying undirected graph, which is the undirected graph obtained by ignoring the directions of the edges of the directed graph. Counting Paths between Vertices • We can use the adjacency matrix of a graph to find the number of paths between two vertices in the graph. • Theorem ○ Let G be a graph with adjacency matrix A with respect to the ordering v_1,…,v_n of vertices ○ (with directed or undirected edges, multiple edges and loops allowed). ○ The number of different paths of length r from v_i to v_j, where r 0 is a positive integer, equals the (i,j)th entry of A^r. • Proof by mathematical induction: ○ Basis Step § By definition of the adjacency matrix, the number of paths from v_i to v_j of length 1 is the (i,j)th entry of A. ○ Inductive Step § For the inductive hypothesis, we assume that that the (i,j)th entry of A^r is the number of different paths of length r from v_i to v_j. § Because A^(r+1)=A^r A, the (i,j)th entry of A^(r+1) equals b_i1 a_1j+b_i2 a_2j+…+b_in a_nj, where b_ik is the (i,k)th entry of A^r. § By the inductive hypothesis, b_ik is the number of paths of length r from v_i to v_k. § A path of length r + 1 from v_i to v_j is made up of a path of length r from v_i to some v_k, and an edge from v_k to v_j. § By the product rule for counting, the number of such paths is the product of the number of paths of length r from v_i to v_k (i.e., b_ik ) and the number of edges from from v_k to v_j (i.e, akj). § The sum over all possible intermediate vertices v_k is b_i1 a_1j+b_i2 a_2j+…+b_in a_nj.

10.3 Representing Graphs and Graph Isomorphism

$Representing Graphs: Adjacency Lists • Definition ○ An adjacency list can be used to represent a graph with no multiple edges by specifying the vertices that are adjacent to each vertex of the graph. • Example 1 • Example 2 Representation of Graphs: Adjacency Matrices • Definition ○ Suppose that G=(V,E) is a simple graph where |V|=n. ○ Arbitrarily list the vertices of G as v_1,v_2,…,v_n. ○ The adjacency matrix A_G of G, with respect to the listing of vertices, is the n×n zero-one matrix with 1 as its (i, j)th entry when v_i and v_j are adjacent, and 0 as its (i, j)th entry when they are not adjacent. ○ In other words, if the graphs adjacency matrix is A_G=[a_ij ], then ○ a_ij={■8(1&if {v_i,v_j } is an edge of G@0&otherwise)┤ • Example 1 ○ The ordering of vertices is a, b, c, d. • Example 2 ○ The ordering of vertices is a, b, c, d. • Note ○ The adjacency matrix of a simple graph is symmetric, i.e., a_ij=a_ji ○ Also, since there are no loops, each diagonal entry a_ii for i = 1, 2, 3, …, n, is 0. • Note ○ When a graph is sparse, that is, it has few edges relatively to the total number of possible edges, it is much more efficient to represent the graph using an adjacency list than an adjacency matrix. ○ But for a dense graph, which includes a high percentage of possible edges, an adjacency matrix is preferable. Adjacency Matrices: Graphs with Loops and Multiple Edges • Adjacency matrices can also be used to represent graphs with loops and multiple edges. • A loop at the vertex v_i is represented by a 1 at the (i, j)th position of the matrix. • When multiple edges connect the same pair of vertices v_i and v_j, (or if multiple loops are present at the same vertex), the (i, j)th entry equals the number of edges connecting the pair of vertices. • Example ○ We give the adjacency matrix of the pseudograph shown here using the ordering of vertices a, b, c, d. Adjacency Matrices: Directed graphs • Adjacency matrices can also be used to represent directed graphs. • The matrix for a directed graph G=(V, E) has a 1 in its (i, j)th position if there is an edge from v_i to v_j, where v_1,v_2,…,v_n is a list of the vertices. • In other words, if the graphs adjacency matrix is A_G=[a_ij ], then ○ a_ij={■8(1&if (v_i,v_j ) is an edge of G@0&otherwise)┤ • The adjacency matrix for a directed graph does not have to be symmetric, because there may not be an edge from v_i to v_j, when there is an edge from v_j to v_i. • To represent directed multigraphs, the value of a_ij is the number of edges connecting v_i to v_j. Representation of Graphs: Incidence Matrices • Definition ○ Let G=(V, E) be an undirected graph with vertices v_1,v_2,…,v_n and edges e_1,e_2,…,e_m. ○ The incidence matrix with respect to the ordering of V and E is the n×m matrix ○ M=[m_ij ],where m_ij={■8(1&when edge e_j is incident with v_i@0&otherwise)┤ • Example ○ Simple Graph and Incidence Matrix ○ ○ The rows going from top to bottom represent v_1 through v_5 ○ The columns going from left to right represent e_1 through e_6. • Example ○ The rows going from top to bottom represent v_1 through v_5 ○ The columns going from left to right represent e_1 through e_8. Isomorphism of Graphs • Definition ○ The simple graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2) are isomorphic if § there is a one-to-one and onto function f from V_1 to V_2 with the property that § a and b are adjacent in G_1 if and only if f(a) and f(b) are adjacent in G_2, for all a and b in V_1. ○ Such a function f is called an isomorphism. ○ Two simple graphs that are not isomorphic are called nonisomorphic. • Example ○ Show that the graphs G=(V, E) and H=(W, F) are isomorphic. • Solution ○ The function f with f(u_1) = v_1, f(u_2)=v_4, f(u_3) = v_3, and f(u_4) = v_2 is a one-to-one correspondence between V and W. ○ Note that adjacent vertices in G are u_1 and u_2, u_1 and u_3, u_2 and u_4, and u_3 and u_4. ○ Each of the pairs f(u_1) = v_1 and f(u_2) = v_4, f(u_1)=v_1 and f(u_3)=v_3 , f(u_2)=v_4 f(u_4)=v_2, f(u_3) = v_3 and f(u_4) = v_2 consists of two adjacent vertices in H • Note ○ It is difficult to determine whether two simple graphs are isomorphic using brute force because there are n! possible one-to-one correspondences between the vertex sets of two simple graphs with n vertices. ○ The best algorithms for determining whether two graphs are isomorphic have exponential worst case complexity in terms of the number of vertices of the graphs. ○ Sometimes it is not hard to show that two graphs are not isomorphic. ○ We can do so by finding a property, preserved by isomorphism, that only one of the two graphs has. ○ Such a property is called graph invariant. ○ There are many different useful graph invariants that can be used to distinguish nonisomorphic graphs, such as the number of vertices, number of edges, and degree sequence (list of the degrees of the vertices in nonincreasing order). ○ We will encounter others in later sections of this chapter. • Example ○ Determine whether these two graphs are isomorphic • Solution ○ Both graphs have eight vertices and ten edges. ○ They also both have four vertices of degree two and four of degree three. ○ However, G and H are not isomorphic. ○ Note that since deg(a) = 2 in G, a must correspond to t, u, x, or y in H, ○ because these are the vertices of degree 2. ○ But each of these vertices is adjacent to another vertex of degree two in H ○ which is not true for a in G. ○ Alternatively, note that the subgraphs of G and H made up of vertices of ○ degree three and the edges connecting them must be isomorphic. ○ But the subgraphs, as shown at the right, are not isomorphic. • Example ○ Determine whether these two graphs are isomorphic. • Solution ○ Both graphs have six vertices and seven edges. ○ They also both have four vertices of degree two and two of degree three. ○ The subgraphs of G and H consisting of all the vertices of degree two and the edges connecting them are isomorphic ○ So, it is reasonable to try to find an isomorphism f. ○ We define an injection f from the vertices of G to the vertices of H that preserves the degree of vertices. ○ We will determine whether it is an isomorphism. ○ The function f with f(u_1)=v_6,f(u_2)=v_3,f(u_3)=v_4,f(u_4)=v_5,f(u_5)=v_1,f(u_6)=v_2 is a one-to-one correspondence between G and H ○ Showing that this correspondence preserves edges is straightforward, so we will omit the details here. ○ Because f is an isomorphism, it follows that G and H are isomorphic graphs. ○ See the text for an illustration of how adjacency matrices can be used for this verification.$

10.2 Graph Terminology and Special Types of Graphs

$Basic Terminology • Definition 1 ○ Two vertices u, v in an undirected graph G are called adjacent (or neighbors) in G if there is an edge e between u and v. ○ Such an edge e is called incident with the vertices u and v and e is said to connect u and v • Definition 2 ○ The set of all a vertices v of G = (V, E), denoted by N(v), is called the neighborhood of v. ○ If A is a subset of V, we denote by N(A) the set of all vertices in G that are adjacent to at least one vertex in A. So, ○ N(A)=⋃8_(v∈A)▒N(V) • Definition 3 ○ The degree of a vertex in a undirected graph is the number of edges incident with it, except that a loop at a vertex contributes two to the degree of that vertex. ○ The degree of the vertex v is denoted by deg(v). Degrees and Neighborhoods of Vertices What are the degrees and neighborhoods of the vertices in the graphs G and H? • For graph G ○ deg(a)=2,deg(b)=deg(c)=deg(f)=4,deg(d)=1,deg(e)=3,deg(g)=0. ○ N(a)={b,f},N(b)={a,c,e,f},N(c)={b,d,e,f},N(d)={c}, ○ N(e)={b,c,f},N(f)={a,b,c,e},N(g)=∅. • For graph H ○ deg(a)=4,deg(b)=deg(e)=6,deg(c)=1,deg(d)=5. ○ N(a)={b,d,e},N(b)={a,b,c,d,e},N(c)={b},N(d)={a,b,e},N(e)={a,b,d}. Degrees of Vertices • Theorem 1 (Handshaking Theorem) ○ If G=(V,E) is an undirected graph with m edges, then ○ 2m=∑_(v∈V)▒deg⁡v • Proof ○ Each edge contributes twice to the degree count of all vertices. ○ Hence, both the left-hand and right-hand sides of this equation equal twice the number of edges. ○ Think about the graph where vertices represent the people at a party and an edge connects two people who have shaken hands. • Example ○ How many edges are there in a graph with 10 vertices of degree six? • Solution ○ Because the sum of the degrees of the vertices is 6⋅10=60, the handshaking theorem tells us that 2m = 60. ○ So the number of edges m = 30. • Example ○ If a graph has 5 vertices, can each vertex have degree 3? • Solution ○ This is not possible by the handshaking theorem ○ Because the sum of the degrees of the vertices 3⋅5=15 is odd. • Theorem 2 ○ An undirected graph has an even number of vertices of odd degree. • Proof ○ Let V_1 be the vertices of even degree and V_2 be the vertices of odd degree in an undirected graph G=(V, E) with m edges.Then § 2m=∑_(v∈V)▒deg⁡v =∑_(v∈V_1)▒deg⁡v +∑_(v∈V_2)▒deg⁡v , where § ∑_(v∈V_1)▒deg⁡v must be even since deg⁡v is even for each v∈V_1 § ∑_(v∈V_2)▒deg⁡v must be even because 2m is even § and the sum of the degrees of the vertices of even degrees is also even. ○ Because this is the sum of the degrees of all vertices of odd degree in the graph, there must be an even number of such vertices Directed Graphs • Definition ○ An directed graph G = (V, E) consists of § V, a nonempty set of vertices (or nodes), and § E, a set of directed edges or arcs. ○ Each edge is an ordered pair of vertices. ○ The directed edge (u,v) is said to start at u and end at v. • Definition ○ Let (u,v) be an edge in G, then § u is the initial vertex of this edge and is adjacent to v and § v is the terminal (or end) vertex of this edge and is adjacent from u. ○ The initial and terminal vertices of a loop are the same. • Definition: ○ The in-degree of a vertex v, denoted deg^−⁡(v), is the number of edges which terminate at v. ○ The out-degree of v, denoted deg^+⁡(v), is the number of edges with v as their initial vertex. ○ Note that a loop at a vertex contributes 1 to both the in-degree and the out-degree of the vertex. • Example ○ In the graph G we have § deg^−⁡(a) = 2, deg^−⁡(b) = 2, deg^−⁡(c) = 3, deg^−⁡(d) = 2, deg^−⁡(e) = 3, deg^−⁡(f) = 0. § deg^+⁡(a) = 4, deg^+⁡(b) = 1, deg^+⁡(c) = 2, deg^+⁡(d) = 2, deg^+⁡(e) = 3, deg^+⁡(f) = 0. • Theorem 3 ○ Let G=(V, E) be a graph with directed edges. Then: § |E|=∑_(v∈V)▒〖deg^−⁡(v)+deg^+⁡(v) 〗 • Proof ○ The first sum counts the number of outgoing edges over all vertices and the second sum counts the number of incoming edges over all vertices ○ It follows that both sums equal the number of edges in the graph. Special Types of Simple Graphs: Complete Graphs • A complete graph on n vertices, denoted by K_n, is the simple graph that contains exactly one edge between each pair of distinct vertices. Special Types of Simple Graphs: Cycles and Wheels • A cycle C_n for n ≥ 3 consists of ○ n vertices v_1,v_2,…,v_n, and ○ edges {v_1,v_2 },{v_2,v_3 },…,{v_(n−1),v_n },{v_n,v_1 } • A wheel W_n is obtained by ○ adding an additional vertex to a cycle C_n for n ≥ 3, and ○ connecting this new vertex to each of the n vertices in C_n by new edges. Special Types of Simple Graphs: n-Cubes • An n-dimensional hypercube, or n-cube, Q_n, is ○ a graph with 2n vertices representing all bit strings of length n, where ○ there is an edge between two vertices that differ in exactly one bit position. Bipartite Graphs • Definition ○ A simple graph G is bipartite if V can be partitioned into two disjoint subsets V_1 and V_2 such that every edge connects a vertex in V_1 and a vertex in V_2 ○ In other words, there are no edges which connect two vertices in V_1 or in V_2. • It is not hard to show that an equivalent definition of a bipartite graph is a graph where it is possible to color the vertices red or blue so that no two adjacent vertices are the same color. Bipartite Graphs and Matchings • Bipartite graphs are used to model applications that involve matching the elements of one set to elements in another, for example: • Job assignments - vertices represent the jobs and the employees, edges link employees with those jobs they have been trained to do. • A common goal is to match jobs to employees so that the most jobs are done. Bipartite Graphs (continued) • Example ○ Show that C_6 is bipartite. • Solution ○ We can partition the vertex set into V_1={v_1, v_3, v_5} and V_2={v_2, v_4, v_6} so that every edge of C_6 connects a vertex in V_1 and V_2. • Example ○ Show that C_2 is not bipartite. • Solution ○ If we divide the vertex set of C_3 into two nonempty sets, one of the two must contain two vertices ○ But in C_3 every vertex is connected to every other vertex ○ Therefore, the two vertices in the same partition are connected. ○ Hence, C_3 is not bipartite. Complete Bipartite Graphs • Definition ○ A complete bipartite graph K_(m,n) is a graph that has its vertex set partitioned into two subsets V_1 of size m and V_2 of size n such that there is an edge from every vertex in V_1 to every vertex in V_2. • Example ○ We display four complete bipartite graphs here. New Graphs from Old • Definition ○ A subgraph of a graph G = (V,E) is a graph (W,F), where W⊂V and F⊂E. ○ A subgraph H of G is a proper subgraph of G if H≠G. • Example ○ Here we show K_5 and one of its subgraphs. • Definition ○ Let G=(V, E) be a simple graph. ○ The subgraph induced by a subset W of the vertex set V is the graph (W,F), where the edge set F contains an edge in E if and only if both endpoints are in W. • Example ○ Here we show K_5 and the subgraph induced by W={a,b,c,e}. • Definition ○ The union of two simple graphs G_1=(V_1, E_1) and G_2=(V_2, E_2) is the simple graph with vertex set V_1∪V_2 and edge set E_1∩E_2 ○ The union of G_1 and G_2 is denoted by G_1∪G_2. • Example$

10.1 Graphs and Graph Models

Graphs • Definition ○ A graph G = (V, E) consists of § a nonempty set V of vertices (or nodes) § and a set E of edges. ○ Each edge has either one or two vertices associated with it, called its endpoints. ○ An edge is said to connect its endpoints. • Example ○ This is a graph with four vertices and five edges. • Remarks ○ The graphs we study here are unrelated to graphs of functions studied in Chapter 2. ○ We have a lot of freedom when we draw a picture of a graph. ○ All that matters is the connections made by the edges, not the particular geometry depicted. ○ For example, the lengths of edges, whether edges cross, how vertices are depicted, and so on, do not matter ○ A graph with an infinite vertex set is called an infinite graph. ○ A graph with a finite vertex set is called a finite graph. We (following the text) restrict our attention to finite graphs. Graph Terminology • In a simple graph each edge connects two different vertices and no two edges connect the same pair of vertices. • Multigraphs may have multiple edges connecting the same two vertices. • When m different edges connect the vertices u and v, we say that {u,v} is an edge of multiplicity m. • An edge that connects a vertex to itself is called a loop. • A pseudograph may include loops, as well as multiple edges connecting the same pair of vertices. • Example ○ This pseudograph has both multiple edges and a loop. Directed Graphs • Definition: ○ An directed graph (or digraph) G = (V, E) consists of § a nonempty set V of vertices (or nodes) § and a set E of directed edges (or arcs). ○ Each edge is associated with an ordered pair of vertices. ○ The directed edge associated with the ordered pair (u,v) is said to start at u and end at v. • Remark: ○ Graphs where the end points of an edge are not ordered are said to be undirected graphs. Some Terminology (continued) • Definition ○ A simple directed graph has no loops and no multiple edges. • Example ○ This is a directed graph with three vertices and four edges. • Definition ○ A directed multigraph may have multiple directed edges. ○ When there are m directed edges from the vertex u to the vertex v, ○ We say that (u,v) is an edge of multiplicity m. • Example ○ In this directed multigraph the multiplicity of (a,b) is 1 and the multiplicity of (b,c) is 2. Graph Models: Computer Networks • When we build a graph model, we use the appropriate type of graph to capture the important features of the application. • We illustrate this process using graph models of different types of computer networks. • In all these graph models, the vertices represent data centers and the edges represent communication links. • To model a computer network where we are only concerned whether two data centers are connected by a communications link, we use a simple graph. • This is the appropriate type of graph when we only care whether two data centers are directly linked (and not how many links there may be) and all communications links work in both directions. • To model a computer network where we care about the number of links between data centers, we use a multigraph. • To model a computer network with diagnostic links at data centers, we use a pseudograph, as loops are needed. • To model a network with multiple one-way links, we use a directed multigraph. • Note that we could use a directed graph without multiple edges if we only care whether there is at least one link from a data center to another data center. Graph Terminology: Summary • To understand the structure of a graph and to build a graph model, we ask these questions: ○ Are the edges of the graph undirected or directed (or both)? ○ If the edges are undirected, are multiple edges present that connect the same pair of vertices? ○ If the edges are directed, are multiple directed edges present? ○ Are loops present? Other Applications of Graphs • We will illustrate how graph theory can be used in models of: ○ Social networks ○ Communications networks ○ Information networks ○ Software design ○ Transportation networks ○ Biological networks • It’s a challenge to find a subject to which graph theory has not yet been applied. • Can you find an area without applications of graph theory? Examples of Collaboration Graphs • The Hollywood graph models the collaboration of actors in films. ○ We represent actors by vertices and we connect two vertices if the actors they represent have appeared in the same movie. ○ We will study the Hollywood Graph in Section 10.4 when we discuss Kevin Bacon numbers. • An academic collaboration graph models the collaboration of researchers who have jointly written a paper in a particular subject. ○ We represent researchers in a particular academic discipline using vertices. ○ We connect the vertices representing two researchers in this discipline if they are coauthors of a paper. ○ We will study the academic collaboration graph for mathematicians when we discuss Erdős numbers in Section 10.4. Transportation Graphs • Graph models are extensively used in the study of transportation networks. • Airline networks can be modeled using directed multigraphs where ○ airports are represented by vertices ○ each flight is represented by a directed edge from the vertex representing the departure airport to the vertex representing the destination airport • Road networks can be modeled using graphs where ○ vertices represent intersections and edges represent roads. ○ undirected edges represent two-way roads and directed edges represent one-way roads. Biological Applications • Graph models are used extensively in many areas of the biological science. • We will describe two such models, one to ecology and the other to molecular biology. • Niche overlap graphs model competition between species in an ecosystem • Vertices represent species and an edge connects two vertices when they represent species who compete for food resources. • Example: This is the niche overlap graph for a forest ecosystem with nine species. • We can model the interaction of proteins in a cell using a protein interaction network. • In a protein interaction graph, vertices represent proteins and vertices are connected by an edge if the proteins they represent interact. • Protein interaction graphs can be huge and can contain more than 100,000 vertices, each representing a different protein, and more than 1,000,000 edges, each representing an interaction between proteins • Protein interaction graphs are often split into smaller graphs, called modules, which represent the interactions between proteins involved in a particular function. • Example: This is a module of the protein interaction graph of proteins that degrade RNA in a human cell.

9.6 Partial Orderings

$Partial Orderings • Definition 1 ○ A relation R on a set S is called a partial ordering, or partial order, if it is reflexive, antisymmetric, and transitive. ○ A set together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R). ○ Members of S are called elements of the poset. • Example 1 ○ Show that the “greater than or equal” relation (≥) is a partial ordering on the set of integers. ○ Reflexivity: a≥a for every integer a. ○ Antisymmetry: If a≥b and b≥a , then a=b. ○ Transitivity: If a≥b and b≥c , then a≥c. • Example 2 ○ Show that the divisibility relation (∣) is a partial ordering on the set of integers. ○ Reflexivity § a∣a for all integers a. (see Example 9 in Section 9.1) ○ Antisymmetry § If a and b are positive integers with a | b and b | a, then a = b § (see Example 12 in Section 9.1) ○ Transitivity § Suppose that a divides b and b divides c. § Then there are positive integers k and l such that b = ak and c = bl § Hence, c=a(kl), so a divides c. § Therefore, the relation is transitive. ○ (Z^+, ∣) is a poset. • Example 3 ○ Show that the inclusion relation (⊆) is a partial ordering on the power set of a set S. ○ Reflexivity: A⊆A whenever A is a subset of S. ○ Antisymmetry: If A and B are positive integers with A⊆B and B⊆A, then A=B. ○ Transitivity: If A⊆B and B⊆C, then A⊆C. Comparability • Definition 2 ○ The elements a and b of a poset (S,≼) are comparable if either a ≼ b or b ≼ a. ○ a,b∈S so that neither a ≼ b nor b ≼ a, then a and b are called incomparable. • Definition 3 ○ If (S,≼) is a poset and every two elements of S are comparable ○ Then S is called a totally ordered or linearly ordered set ○ And ≼ is called a total order or a linear order. ○ A totally ordered set is also called a chain. • Definition 4 ○ (S,≼) is a well-ordered set if it is a poset such that § ≼ is a total ordering § every nonempty subset of S has a least element. Hasse Diagrams • A Hasse diagram is a visual representation of a partial ordering that leaves out edges that must be present because of the reflexive and transitive properties. • A partial ordering is shown in (a) of the figure above. • The loops due to the reflexive property are deleted in (b). • The edges that must be present due to the transitive property are deleted in (c). • The Hasse diagram for the partial ordering (a), is depicted in © Lexicographic Order • Definition ○ Given two posets (A_1,≼_1) and (A_2,≼_2) ○ The lexicographic ordering on A_1⨉A_2 is defined by specifying that ○ (a_1, a_2) is less than (b1,b2), that is, (a_1, a_2) ≺ (b_1,b_2), ○ either if a_1 ≺_1 b_1 or if a_1=b_1 and a_2 ≺_2 b_2. ○ This definition can be easily extended to a lexicographic ordering on strings (see text). • Example ○ Consider strings of lowercase English letters ○ A lexicographic ordering can be defined using the ordering of the letters in the alphabet ○ This is the same ordering as that used in dictionaries. ○ discreet ≺ discrete, because these strings differ in the seventh position and e ≺ t. ○ discreet ≺ discreetness, because the first eight letters agree, but the second string is longer. Well Ordered Induction • Theorem ○ If (S,≤) is a well ordered poset and P is a property s.t. ○ If P(y) is true for all y x, then P(x) is true ○ Then P is true for all elements in the poset • Example ○ Suppose that a_(m,n) is defined for (m,n)∈N×N § a_0,0=0 § a_(m,n)={■8(a_(m−1,n)+1&if n=0 and m 0@a_(m,n−1)+n&if n 0)┤ ○ Show that a_(m,n)=m+n(n+1)/2 is defined for all (m,n)∈N×N • Solution ○ Use induction ○ Basis Step § a_0,0=0+(0⋅1)/2 ○ Inductive Step § Assume that a_(m^′,n^′ )=m^′+(n^′ (n^′+1))/2 whenever § (m^′,n′) is less than (m,n) in the lexicographic ordering of N×N § If n=0, by the inductive hypothesis, we can conclude □ a_(m,n)=a_(m−1,n)+1=m−1+n(n+1)/2+1=m+n(n+1)/2 § If n 0, by the inductive hypothesis, we can conclude □ a_(m,n)=a_(m−1,n)+1=m+n(n−1)/2+n=m+n(n+1)/2 Maximal and Minimal Elements • Definition ○ If (S,≤) is a ordered poset then an element a is ○ Minimal if there is no element b s.t. b≤a, and b is not equal to a ○ Maximal if there is no element b s.t. a≤b, and b is not equal to a ○ Greatest if for every element b∈S, b≤a ○ Least if for every element b∈S, a≤b • Intuition ○ Minimal: No element is strictly smaller ○ Least: Every other element is bigger ○ Maximal: No element is strictly bigger ○ Greatest: Every other element is smaller • Example 1 ○ Let {2,4,5,10,12,20,25} ordered by divisibility relation ○ Which element of {2,4,5,10,12,20,25} are maximal and which are minimal? ○ Is there a least or greatest element? ○ Maximal: 12, 20, 25 ○ Minimal: 2,5 ○ No least or greatest element. • Example 2 ○ Given an example of a poset with no minimal element and two maximal elements? ○ {〖1,1〗^∗,0,−1,−2,−3,…} where 1≥0≥−1≥−2≥…, and 1^∗≥0≥−1≥−2≥… Topological Sorting • Definition ○ If (S,R) is a partial ordering and (S,R^∗ ) is a total ordering then we say that ○ The two are compatible if R is a subset of R^∗ • Theorem ○ Let S be a finite set ○ Every partial order R on S has a compatible total order R^∗ • Proof ○ Every nonempty finite partial order has a minimal element ○ This can be proved by induction on the size of S ○ We should build a total ordering of S be selecting § a_1 to be a minimal element of S § a_2 to be the minimal element of S−{a_1 } § a_i to be the minimal element of S−{a_1,a_2,…,a_(i−1) } § a_n to be the least available element of S ○ We set R^∗={(a_i,a_i )|j≥i}$

Shawn Zhong

Shawn Zhong

10.4 Connectivity

10.3 Representing Graphs and Graph Isomorphism

10.2 Graph Terminology and Special Types of Graphs

10.1 Graphs and Graph Models

9.6 Partial Orderings

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