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Chapter 13: Problem 11

\([\mathrm{BB}]\) Faced with the undeniable fact that his town's street system does indeed have some bridges, Mayor Murphy decides that all bridges should remain two-way streets, but all other streets should be made one-way. Can this be done so as to allow (legal) travel from any intersection to any other?

Short Answer

Expert verified
Yes, by making non-bridge streets one-way while maintaining strong connectivity.

Step by step solution

01

Understanding Graph Terminology

To solve this problem, we must first understand what the question implies. The town's street system can be represented as a graph where intersections are vertices and streets are edges. Bridges represent edges that are part of some cycle in the graph.
02

Identify Bridges

Bridges in a graph are edges that, if removed, would increase the number of connected components of the graph. We must identify these bridges since they need to remain two-way streets.
03

Convert Non-Bridge Streets to One-Way

For all other streets that are not bridges, we will convert them into one-way streets. This will make our task possible without disconnecting the graph. The one-way assignments should allow strong connectivity.
04

Ensure Strong Connectivity

For the resulting directed graph to allow travel from any intersection to any other, it needs to be strongly connected. Ensuring this implies having a directed path from any vertex to any other.
05

Utilize Strongly Connected Components

Identify the strongly connected components of the original undirected graph. Utilize algorithms like Tarjan's or Kosaraju's to establish them, maintaining connectivity through non-bridge one-way streets.
06

Verify Solution

Ensure that all bridges remain two-way and that every non-bridge street forms part of a directed path that allows travel between any pair of vertices. This criterion certified by the proposal guarantees solution correctness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bridges in Graphs
In graph theory, a bridge, also known as a cut-edge, is an edge that, if removed, increases the number of connected components in a graph. Imagine a bridge as a critical link that holds two portions of a graph together. Without it, the graph would split into separate, disconnected segments.
This concept is crucial, as bridges play a vital role in maintaining the integrity and connectivity of a graph.
  • Bridges are important because they can signify a point of vulnerability in a network.
  • In practical terms, think of a bridge in a graph as a bottleneck in road systems or communication networks that is essential to maintaining flow.
Detecting bridges is the first step in problems where connectivity needs to be ensured even when certain connections go one way, as with converting roads to one-way streets except bridges.
Strongly Connected Components
Strongly connected components are a key concept in the world of directed graphs. They refer to subgraphs where there is a directed path from any vertex to every other vertex within the subgraph. In essence, these components can be thought of as loops of connectivity.
Understanding strongly connected components is crucial for ensuring all vertices in a graph can be reached, a requirement in systems transforming most connections into one-way links.
  • Each strongly connected component can be traveled completely within if you start from any point in it.
  • Algorithms like Tarjan's or Kosaraju's can efficiently find strongly connected components in a graph.
By finding these components, we assure that the converted graph remains navigable in all directions within each component.
Directed Graphs
A directed graph, or digraph, is a set of vertices connected by edges, where the edges have a direction. Think of it as a one-way street map, where you can only travel in a designated direction on each edge.
In directed graphs, ensuring connectivity means carefully arranging edges so that every node is accessible following the directions laid out by the edges.
  • Directed edges prevent the arbitrary traversal of connections, making connectivity through directed paths a key design goal.
  • Converting a town's street system to a directed graph involves careful planning to maintain accessibility.
Directed graphs demand thoughtful edge direction assignments to achieve full connectivity, especially when non-essential edges become one-way.
Graph Connectivity
Graph connectivity refers to the level of connection between various vertices in a graph. It indicates how difficult it is for the graph to become disconnected if edges or vertices are removed.
High connectivity means that there are numerous paths between vertices, making it less likely that the graph will fracture if some connections are altered.
  • Connected graphs have at least one path between any pair of vertices. This is essential for ensuring robust networks.
  • Strong connectivity in directed graphs involves both reaching and being reachable from every vertex, requiring careful planning of edges.
Maintaining graph connectivity is crucial in redesigning road systems or networks, ensuring seamless travel across all points even when some paths become one-way.

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Most popular questions from this chapter

\([\mathrm{BB}]\) Let \(\mathcal{G}\) be a graph with a strongly connected orientation assigned to it. Suppose that the direction of thearrow on each edge of \(\mathcal{G}\) is reversed. Is the new digraph still strongly connected? Explain.

How many bridges does a tree with \(n\) vertices have? Explain.

\([\mathrm{BB} ;(\rightarrow)]\) Prove that an edge \(e\) in a connected graph \(\mathcal{G}\) is a bridge if and only if there are vertices \(u\) and \(v\) in \(\mathcal{G}\) such that every path from \(u\) to \(v\) requires the edge \(e\).

[BB] Give an example of a connected graph \(\mathcal{G}\) which has a spanning tree not obtainable as a depth-first spanning tree for \(\mathcal{G}\).

(a) \([\mathrm{BB}]\) Let \(v\) be a vertex in a graph \(\mathcal{G}\) which is labeled by the depth-first search algorithm. Prove that the algorithm labels all the vertices adjacent to \(v .\) (b) Prove that if \(\mathcal{G}\) is connected, the depth-first algorithm labels every vertex of \(\mathcal{G}\).
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