# Prove the decomposition of a graph w/ even edges produce a 2-path set

• Superyoshiom
In summary, the conversation discusses using a graph with three vertices and two edges as a base case and decomposing it for a path length of 2. The inductive step involves removing an edge to create two parts of the graph, one with odd edges and one with even edges. The question asks if the subgraph with even edges can be decomposed to achieve a path length of 2. The type of graph decomposition being referred to is not specified.
Superyoshiom
Homework Statement
Using induction, we need to prove that given a graph G where the number of edges is even, if we decompose it then there is a decomposition will create a set of paths of length 2
Relevant Equations
G = (V,E), where V are the vertices and E are the edges
For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2.

The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two parts of a graph. Assuming the graph has even edges to begin with, removing edges creates one subgraph with odd edges and one with even edges. Can I assume then, that the subgraph with even edges can be decomposed to make path lengths of 2?

There is more than one kind of graph decomposition. Please define the type to be used in this question.
Does it just mean an arbitrary partitioning of the edges?

## 1. What is a graph?

A graph is a mathematical structure consisting of a set of vertices (or nodes) and a set of edges connecting these vertices. It is used to represent relationships between objects or entities.

## 2. What does it mean for a graph to have even edges?

A graph with even edges means that all of its vertices have an even number of edges connected to them. In other words, each vertex has an even degree.

## 3. What is decomposition of a graph?

Decomposition of a graph refers to breaking down a larger graph into smaller subgraphs. This is often done to simplify the analysis of the graph or to identify certain patterns or structures within the graph.

## 4. How does a graph with even edges produce a 2-path set?

When a graph with even edges is decomposed, it can be shown that it will always produce a 2-path set. This means that the subgraphs formed from the decomposition will consist of two paths connecting the same two vertices.

## 5. Why is it important to prove the decomposition of a graph with even edges produces a 2-path set?

Proving this statement is important because it helps us better understand the properties and structures of graphs. It also has applications in various fields, such as computer science and network analysis, where graphs are used to model real-world systems.

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