/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Show that the block graph of any... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 6

Show that the block graph of any connected graph is a tree.

Short Answer

Expert verified
By contradicting the assumption that a block graph can contain a cycle, it is shown that the block graph of any connected graph must always be a tree.

Step by step solution

01

Define a Block Graph

A block graph of a given graph is a graph whose vertices represent the blocks of the given graph, and two vertices are adjacent if and only if their corresponding blocks share a vertex in the given graph.
02

Assume a Contradiction

Assume to the contrary, that there exists a graph such that its block graph is not a tree. Thus, it must contain a cycle.
03

Show Contradiction

Since each vertex in a block graph is a maximal connected subgraph without any cut-vertex, two adjacent vertices (blocks) in the block graph must share a common vertex (the cut vertex). In the cycle in the block graph, the cut vertices shared by the adjacent vertices must form a cycle in the original graph, which contradicts our assertion since a cycle is a block on its own.
04

Finalize the Proof

As we have reached a contradiction, our assumption that the block graph of a connected graph can contain a cycle must be incorrect. Therefore, the block graph of any connected graph is a tree.

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Key Concepts

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

Understanding Connected Graphs
In the world of graph theory, a connected graph plays a fundamental role. A graph consists of vertices (or nodes) and edges connecting these vertices. In a connected graph, you can always find a path from any vertex to any other vertex in the graph. This means there's only one piece, so to speak, linking every vertex together.
Key characteristics of connected graphs include:
  • There is a sequence of vertices known as a path connecting any pair of vertices.
  • Removing any edge does not disconnect the graph.
  • They form the basis for studying other types of graphs, like block graphs and trees.
Connected graphs are crucial for modeling networks, whether it's computer networks, social networks, or other types of linked structures.
Defining a Tree in Graph Theory
A tree is a specific type of graph that is both simple and incredibly useful. Essentially, a tree is a connected graph with no cycles.
Some important features of trees include:
  • There is exactly one path between any two vertices, which ensures no cycles.
  • Removing any edge will make the graph disconnected, highlighting the importance of each link.
  • The number of edges is always one less than the number of vertices ( \(E = V - 1\)).
Trees are often used to represent hierarchical structures, such as organizational charts or file systems. They simplify complex relationships by ensuring a clear path from the root to any other node.
Exploring Cycles in Graphs
A cycle in a graph is a path that starts and ends at the same vertex, forming a loop. While cycles can be interesting, they are absent in trees and other special graphs like block graphs.
Characteristics of cycles include:
  • They contain at least three vertices and edges arranged in a closed loop.
  • Removing an edge in a cycle does not affect graph connectivity, which differentiates them from trees.
  • Cycles can make a network more resilient by providing alternative paths.
Understanding cycles is essential in network design to possibly avoid redundancies or plan fail-safes through redundant paths.
What is a Block in Graph Theory?
Blocks are substructures within a graph that offer incredible insights. A block is a maximal connected subgraph that has no cut-vertex. This means it's a portion of the graph that stands strong even if you remove any one vertex within it.
Key features of blocks include:
  • They contain no vertices (cut-vertices) whose removal would disconnect the block.
  • Each block is connected, and any two blocks in a graph share at most one vertex.
  • Blocks help understand the strength and resilience of a network. Where they exist, the network can sustain vertex failures without falling apart.
Blocks help in analyzing the robustness of networks and play a critical role in understanding the broader structure of graphs and their connections through block graphs.

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Most popular questions from this chapter

Let \(G\) be a 3-connected graph, and let \(x y\) be an edge of \(G\). Show that \(G / x y\) is 3-connected if and only if \(G-\\{x, y\\}\) is 2-connected.

Show, without using Menger's theorem, that any two vertices of a \(2-\) connected graph lie on a common cycle.

Let \(k \geqslant 2\). Show that every \(k\)-connected graph of order at least \(2 k\) contains a cycle of length at least \(2 k\).

Let \(G\) be a 2-connected graph but not a triangle, and let \(e\) be an edge of \(G .\) Show that either \(G-e\) or \(G / e\) is again 2-connected. Deduce a constructive characterization of the 2-connected graphs analogous to Theorem 3.2.2.

Show that \(k\)-linked graphs are \((2 k-1)\)-connected. Are they even \(2 k\) connected?
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