Read & Interact: Rosen: Chapter 10.1 - 10.4

Match these complete bipartite graphs with the number of edges they have. Instructions

12 10 9 7 8

Match each vertex of this simple graph with the list of vertices adjacent to it.

a matches Choice b, d b matches Choice a, d, e c matches Choice e d matches Choice a, b, e e matches Choice b, c, d

Which of these statements about connectedness in directed graphs are true?

A directed graph is a strongly connected graph if and only if there is a directed path from any vertex to any other vertex. Given any two vertices in a strongly connected graph, it is possible to find a directed path from one vertex to the other. If a directed graph is strongly connected or weakly connected, then the underlying undirected graph is connected.

In the graph of employees and jobs, there is an edge between an employee and a job if that employee has been trained for that job. Hence, the graph is bipartite, with bipartition (V1, V2) where V1 is the set of employees and V2 is the set of jobs. Match each description in the right hand column with the appropriate set of edges in the left hand column.

A set of edges where each edge represents an assignment of employees to jobs matches Choice A matching A set of the largest possible number of edges, each representing an assignment of employees to jobs matches Choice A maximal matching A set of edges where every employee is a vertex in one of these edges matches Choice A complete matching from V1 to V2

Match the graph with the drawing that represents it.

B C D A

Match the names of special families of graphs with their descriptions.

complete graph matches Choice a graph where each vertex is connected to every other vertex by an edge cycle matches Choice a graph constructed by connecting each vertex in a list to the next vertex in the list and connecting the last vertex with the first. wheel matches Choice a graph constructed by adding an additional vertex to a cycle and connecting it to each of the other vertices n-cube matches Choice a graph whose vertices represent the bit strings of length n where two vertices are connected if they represent strings that differ in one bit

Match the statement with the appropriate terminology from graph theory.

degree matches Choice The _____ of a vertex in an undirected graph is the number of edges incident with it. isolated matches Choice A vertex with no neighbors is called _____. pendant matches Choice A vertex with exactly one neighbor is called _____.

Two simple graphs are _______ if there is a bijection from the vertices of the first graph to the vertices of the second such that two vertices are adjacent in the first graph if and only if their images are adjacent in the second.

isomorphic

A _____ in a simple graph is a subset of the set E of edges of the graph so that no two edges are incident with the same vertex.

matching

When edges and vertices are removed from a graph without removing the endpoints of any remaining edges, a smaller graph is obtained, known as a ________ of the original graph.

subgraph

A directed graph is called _____ connected if there is a path between every two vertices in the underlying undirected graph.

Blank 1: weakly or weak

Which of these is the adjacency matrix of this pseudograph

B

Which of these statements about adjacency matrices are always true?

All the entries on the main diagonal of the adjacency matrix of a simple graph are 0s. The adjacency matrix of a simple graph with n vertices is an n × n matrix. The adjacency matrix of a simple graph is always symmetric.

Which of these applications can be modeled by a matching?

Assigning jobs to employees so that no two employees do the same job and no employee has more than one job Assigning students to chairs in a classroom so that no student has two chairs and no chair has two students

Which of these applications can be modeled by a matching?

Assigning students to chairs in a classroom so that no student has two chairs and no chair has two students Assigning jobs to employees so that no two employees do the same job and no employee has more than one job

An undirected graph is called _____ if there is a path between every pair of distinct vertices of the graph. Otherwise, this graph is called _____.

Blank 1: connected Blank 2: disconnected or not connected

A vertex that when it and all its incident edges are removed from a graph produces a subgraph with more connected components than the original graph is called a _____ vertex. An edge whose removal produces a graph with more connected components than in the original graph is called a _____ edge.

Blank 1: cut or articulation Blank 2: cut or bridge

A ______ in an undirected graph is a sequence of edges in which each edge after the first shares an endpoint with the previous edge in the sequence. A ______ is such a sequence that begins and ends at the same vertex.

Blank 1: path or walk Blank 2: circuit, closed walk, or cycle

Match the graph with the drawing that represents it.

C4�4 matches Choice Graph B K4�4 matches Choice Graph C W4�4 matches Choice Graph D Q2�2 matches Choice Graph A

Match the description of the vertex with its degree.

Degree 0 matches Choice Vertex of a person in the friendship graph who has no friends Degree 10 matches Choice Vertex of an actor in the Hollywood graph who has been in just one movie with ten other actors Degree 1 matches Choice Vertex of a mathematician in the collaboration graph representing jointly published papers who has exactly one coauthor Degree 8 matches Choice Vertex of a species that competes with all other species in a niche overlap graph representing competition between nine species

Which of the graphs are connected?

G3�3 G2

Which of these statements about subgraphs are true?

If G = (V , E), the graph G - e obtained by deleting the edge e from G has vertex set V and edge set E - {e}. If G1 = (V1 , E1) and G2 = (V2 , E2), the graph union G1 ∪ G2 has vertex set V1 ∪ V2 and edge set E1 ∪ E2. If G = (V , E), the graph G + e obtained by adding an edge e with endpoints in V has vertex set V and edge set E ∪ {e}.

Which of these statements are true regarding algorithms for graph isomorphism?

It is not known whether it is possible for a tractable algorithm to be found that can determine whether two graphs are isomorphic. We can run software today that can determine whether two graphs with 100 vertices are isomorphic in a reasonable amount of time. Algorithms that have linear average-case time complexity have been found for determining whether two graphs are isomorphic.

Match the topology for a local area network with the appropriate type of graph that can model this topology.

K C W

Put the steps in the correct order to construct an induction proof that the entries of the rth power of the adjacency matrix of a graph give the number of different paths with specified start and end vertices.

Let Assume Since A Summing The

Construct a proof that there is a simple path between every pair of distinct vertices of a connected undirected graph

Let G For Deleting This Hence

Construct a proof that there is a simple path between every pair of distinct vertices of a connected undirected graph.

Let G For Deleting This Hence

Match each vertex sequence in this undirected graph with its description in terms of paths and circuits.

Path of length 4 matches Choice e,c,b,a,d Path of length 5 matches Choice b,a,d,b,e,c Circuit of length 5 matches Choice f,c,b,a,e,f Circuit of length 6 matches Choice a,b,c,e,b,d,a Neither a path nor a circuit matches Choice c,f,b,a,e,c

Which of these will show that two graphs are isomorphic?

Showing that it is possible to change the adjacency matrix for one graph into that of the other by interchanging rows and by interchanging columns Finding a one-to-one correspondence between the vertices that preserves the adjacency of the vertices

Match the graphs shown in the figure to the appropriate description.

Strongly connected matches Choice G3�3 Not strongly connected, but weakly connected matches Choice G1�1 Not strongly connected and not weakly connected matches Choice G2�2

Which of these are graph invariants?

The number of vertices in a graph The number of edges in a graph Whether the graph is bipartite The number of vertices of degree n for every integer n

Construct a proof that a simple graph is bipartite if and only if we can color each vertex of the graph either blue or red so that no two adjacent vertices are colored the same by putting the steps provided in correct order.

We Then If To Let The

Match each vertex of this directed graph with the list of the terminal vertices of the edges that have this vertex as initial vertex.

a matches Choice a,b,c b matches Choice c c matches Choice a d matches Choice a,c e matches Choice a,b,d

The adjacency matrix of a graph with vertices a, b, c, and d, in that order, is 00110010110110100011001011011010 Which of these pairs of vertices are connected by an edge in this graph?

a and c c and d b and c

The adjacency matrix of the multigraph G, with vertices a, b, c, d, in that order, is 02102003100103100210200310010310 Match the pairs of vertices with the number of different paths of length 3 between them.

a,a matches Choice 0 b,a matches Choice 31 c,a matches Choice 12 c,d matches Choice 17 d,b matches Choice 44

Match each statement with the appropriate terminology from graph theory.

adjacent - _____ vertices are endpoints of the same edge in an undirected graph. incident - An edge connecting vertices u and v is called _____ with the vertices u and v. degree-The _____ of a vertex is the number of edges that have this vertex as an endpoint. pendant-A vertex is _____ if there is only one edge that has this vertex as an endpoint.

Which of these vertices are cut vertices in this graph?

b f c