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Chapter 14: Problem 46

What is a subgraph?

Short Answer

Expert verified
A subgraph is a graph that consists of some (or all) the vertices and some (or all) the edges of a larger graph, maintaining the same connections.

Step by step solution

01

Definition of a Graph

In the context of graph theory, a graph consists of vertices (or nodes) and edges (or arcs) that connect these vertices. A graph is formally described as \(G = (V, E)\), where \(V\) is the set of vertices and \(E\) is the set of edges.
02

Definition of a Subgraph

A subgraph \(S\) of a graph \(G\) is a graph whose vertices and edges are all in \(G\), but it doesn't have to contain all of the edges or vertices of \(G\). Formally, a subgraph \(S\) of \(G\) is a graph such that all of \(S\)鈥檚 vertices and edges belong to \(G\) and the endpoints of each of \(S\)鈥檚 edges are the same as the endpoints of that edge in \(G\).
03

Example of a Subgraph

Consider a graph \(G\) with vertices \(V = \{a, b, c, d\}\) and edges \(E = \{(a,b), (b,c), (c,d), (d,a)\}\). A subgraph \(S\) of graph \(G\) might be \(S = \{a, b, c\}\) with edges \(E' = \{(a,b), (b,c)\}\).

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