Chapter 10
Which of the statements about connected components are true?
- No vertex can belong to more than one connected component of the graph. - A graph that is not connected has more than one connected component. - A graph is the union of its connected components. - A connected component of a graph is a maximal connected subgraph of the graph.
Match each characterization of a connected planar simple graph in the left hand column with the number of regions such a graph splits in the plane in the right hand column. 20 vertices each of degree 4 25 vertices and 35 edges 10 vertices of degree 3 and 6 vertices of degree 4 30 vertices and 39 edges
20 vertices each of degree 4 --> 22 25 vertices and 35 edges --> 12 10 vertices of degree 3 and 6 vertices of degree 4 --> 13 30 vertices and 39 edges --> 11
Which of these statements about Hamilton circuits are true?
A simple graph with n vertices must have a Hamilton circuit if n ≥ 3 where deg(u) + deg(v) ≥ n whenever u and v are nonadjacent vertices. A simple graph with n vertices must have a Hamilton circuit if n ≥ 3 where the degree of every vertex is at least n / 2. (This is just Ore's and Dirac's theorem)
Which of these problems is solved by determining whether the graph that is plainly implied by the problem is planar or not?
Designing a road network that can be built without using underpasses or overpasses Determining whether we can connect utilities to house at ground level without any connections crossing Determining whether we can build an electronic circuit so that none of the connections of transistors cross
Which of these are typical applications of vertex connectivity or edge connectivity?
Determining the minimum number of intersections in a highway network whose closure makes it impossible to travel between every pair of intersections Finding the minimum number of roads that must be closed to disconnect a road network Finding the minimum number of fiber optic links that disconnect a telephone network when they are down Determining the minimum number of routers that disconnect a computer network when they are out of service
Which of these are applications of Euler paths or circuits?
Finding a route for a postman so that the route covers each street in the route exactly once Finding a route for a snowplow that traverses each street in the route exactly once
Which of these problems involve finding the shortest path in a weighted graph?
Finding the airline route with the least total flight time between two cities with zero or more stops Finding the least expensive airline route with zero or more stops between two cities when the cost is the sum of the cost of the legs Finding the total distance a traveling salesman must drive on a road network to travel between two cities Finding the an airline route between two cities with the shortest total mileage
Which of these criteria can be used to show that K3,3 is nonplanar?
If a connected planar simple graph has e edges and v vertices, where v ≥ 3, and no circuits of length 3, then e ≤ 2v - 4.
Kuratowski's theorem says that a graph is nonplanar if it contains a subgraph homeomorphic to which of these graphs?
K5 K3,3
Which of these graphs have a Hamilton circuit?
K5 , C5, and W4
What is the name for a graph that has a number assigned to each edge?
Weighted graph
Which of these vertices are cut vertices in this graph?
c, b, f
An undirected graph is called _______ if there is a path between every pair of distinct vertices of the graph. Otherwise, this graph is called _______.
connected, disconnected
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.
cut/articulation, cut/bridge
The operation on a simple graph of removing an edge {u, v}, and then adding a new vertex w together with new edges {u, w}, and {w, v}, is called an ________ subdivision. Two graphs are called _________ if they can be obtained from the same graph by a sequence of such operations.
elementary, homeomorphic
A graph is called ______ if it can be drawn in the plane without any edges crossing.
planar
homeomorphic
two undirected graphs are homeomorphic if they can be obtained from the same graph by a sequence of elementary subdivisions
A directed graph is called _______ connected if there is a path between every two vertices in the underlying undirected graph.
weakly
Which of these statements about the vertex connectivity and the edge connectivity of a graph G are true for all graphs G?
λ(G)≤ minv∊V deg(v) κ(G)≤ minv∊V deg(v) If G is disconnected, then κ(G) = 0.
Which of these statements about Hamilton paths and Hamilton circuits are true?
A graph with a pendant vertex cannot have a Hamilton circuit. A Hamilton circuit cannot contain a smaller circuit within it. The complete bipartite graph Kn,n+1 has a Hamilton path for n≥3.
Which of these statements are true about paths in the acquaintanceship graph of all people in the world? - For every acquaintance of yours, you are connected to this person by a path of length 1. - For every acquaintance of yours, the shortest path linking you to this person is of length 2. - Every two people in the world are connected by a path. - Social scientists have conjectured that most pairs of people are connected by a path of length 4 or less.
For every acquaintance of yours, you are connected to this person by a path of length 1. Social scientists have conjectured that most pairs of people are connected by a path of length 4 or less.
A simple path in a graph G passing through all vertices in the graph exactly once is called a _______ path.
Hamilton
Euler's Formula
Let G be a connected planar simple graph with e edges and v vertices. Let r be the number of regions in a planar representation of G. Then r = e - v + 2
Which of these are applications of Hamilton circuits or paths?
Minimizing the size of errors when analog signals are converted to digital through the use of Gray codes Finding the ways a traveling salesman can visit all the cities he services Finding a way for a garbage truck to visit all customers without overlap
Which of these graphs are planar?
Q3 W5 C5
Match the graphs shown in the figure to the appropriate description.
Strongly connected: G3 Not strongly connected, but weakly connected: G1 Not strongly connected and not weakly connected: G2
Which of these provides a correct analysis of the famous Köningsberg bridge problem?
The bridges multigraph has four odd vertices, so it has no Euler circuit. Thus one cannot cross each bridge exactly once and return to the start.
How to find edges from vertices and their degrees
The sum of the degrees of the vertices (multiplying the vertices by the degrees) is equal to twice the number of edges
Which of these statements about planar graphs are true?
There are 10 regions in a planar representation of a graph with 20 edges and 12 vertices. A connected planar simple graph with 15 vertices and 16 edges could be planar.
A subset W of the vertex set V of a connected graph G = (V , E) is a ________ if G - W is disconnected.
Vertex cut Separating Set
Suppose that G is a graph. A set of edges E' with the property that G - E' is disconnected is called an edge ______ of G. The edge _____ of G, denoted by λ(G), is the minimum number of edges in such a set of edges.
cut, connectivity
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.
path/walk, circuit/closed walk/cycle
Which of these relationships about the number of edges e, the number of vertices v, and the number of regions r in a planar graph is correct?
r = e - v + 2
degree of a region
the number of edges on the boundary of this region
Elementary subdivision
the removal of an edge {u, v} of an undirected graph and the addition of a new vertex w together with edges {u, w} and {w, v}