(Continuation of Exercise 23.) Let V1, V2, …, Vk be the different sets that occur among the C(x)

step 1 In this question, we are present with a graph G that has a loop on every vertices. And we define

(Continuation of Exercise 23.) Let V1, V2, …, Vk be the...

(Continuation of Exercise 23.) Let $V_{1}, V_{2}, \ldots, V_{k}$ be the different sets that occur among the $C(x)^{\prime}$ 's. Prove the following: (i) $V_{1}, V_{2}, \ldots, V_{k}$ form a partition of the vertex set $V$ of $G$. (ii) The general subgraphs $G_{1}=\left(V_{1}, E_{1}\right), G_{2}=\left(V_{2}, E_{2}\right), \ldots, G_{k}=\left(V_{k}, E_{k}\right)$ of $G$ induced by $V_{1}, V_{2}, \ldots, V_{k}$, respectively, are connected. The induced subgraphs $G_{1}, G_{2}, \ldots, G_{k}$ are the connected components of $G$.

(Continuation of Exercise 23.) Let $V_{1}, V_{2}, \ldots, V_{k}$ be the different sets that occur among the $C(x)^{\prime}$ 's. Prove the following:
(i) $V_{1}, V_{2}, \ldots, V_{k}$ form a partition of the vertex set $V$ of $G$.
(ii) The general subgraphs $G_{1}=\left(V_{1}, E_{1}\right), G_{2}=\left(V_{2}, E_{2}\right), \ldots, G_{k}=\left(V_{k}, E_{k}\right)$ of
$G$ induced by $V_{1}, V_{2}, \ldots, V_{k}$, respectively, are connected.
The induced subgraphs $G_{1}, G_{2}, \ldots, G_{k}$ are the connected components of $G$.

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Key Concepts

Connected Components

A connected component of a graph is a maximal subgraph in which every pair of vertices is connected by a path. Connected components naturally partition the graph's vertex set because each vertex belongs exactly to one component, confirming that they are disjoint and cover the entire graph.

Induced Subgraph

An induced subgraph is formed from a larger graph by selecting a subset of vertices and taking all edges that connect pairs of vertices within that subset. This concept is crucial in order to study the properties of a subcomponent of a graph, particularly its connectivity, while retaining the original structure between the chosen vertices.

Partition of a Set

In graph theory, a partition of a set refers to the division of the vertex set into mutually exclusive, non-empty subsets such that every vertex appears in exactly one subset. This partition is based on an equivalence relation, in this case defined by the connectedness properties, which ensures that the subsets are disjoint and collectively exhaustive.

Graph Connectivity

Connectivity in a graph means that any two vertices in a subgraph are connected by a path. This fundamental concept is used to ascertain whether a set of vertices forms a cohesive unit, ensuring that there is a sequence of adjacent edges between any two vertices within that set.

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