Question

Prove that the blocks of a tree are all $K_{2}$ 's.

   Prove that the blocks of a tree are all $K_{2}$ 's.

Introductory Combinatorics

Introductory Combinatorics

Richard A. Brualdi 5th Edition

Chapter 12, Problem 60

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Solved by Expert Robin Corrigan

12/31/2020

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Recall that a block is a maximal connected subgraph with no cut vertices. In other words, a block is a subgraph that cannot be disconnected by removing a single vertex. Show more…

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Prove that the blocks of a tree are all $K_{2}$ 's.

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Key Concepts

Tree

A tree is a connected graph that contains no cycles, meaning there is exactly one simple path between any two vertices. This property ensures that the addition or removal of any edge or vertex can quickly affect connectivity, and it is fundamental in understanding why trees decompose into very simple structures.

Block

A block in a graph is a maximal connected subgraph that does not contain any cut vertices. In other words, it is a part of the graph that cannot be split into separate pieces by removing a single vertex. This concept is key to understanding the internal structure of graphs through their connectivity properties.

Cut Vertex

A cut vertex (or articulation point) is a vertex whose removal increases the number of connected components in the graph. In a tree, every non-leaf vertex acts as a cut vertex, except in the simplest case of a single edge, highlighting why the building blocks (or blocks) of a tree can only be edges.

Maximal 2-connected Subgraph

A maximal 2-connected subgraph is one that cannot be expanded by including any adjacent vertex without introducing a cut vertex. Since trees are acyclic and every vertex (except possibly the ends) is a cut vertex, the only subgraphs that do not have any cut vertices in a tree are the trivial ones, namely the individual edges (K?’s).

Graph Connectivity

Graph connectivity relates to how robustly a graph is connected. In the study of blocks, which are defined by their connectivity properties, understanding how a graph breaks apart under vertex removal is crucial. For trees, the inherent lack of cycles means that connectivity is minimal, so the only connected, non-separable parts are the edges.

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