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Today, we're going to discuss subgraphs. Does anyone know what a subgraph is?
Isn't it just a part of some graph?
Exactly! A subgraph consists of a subset of vertices and edges from the original graph. It maintains some structure derived from the parent graph. For example, if G has vertices {A, B, C} and edges {(A, B), (B, C)}, then H can have vertices {A, B} and an edge {(A, B)} from G.
What makes it a 'proper' subgraph?
Great question! A proper subgraph must have at least one vertex or edge that is not present in the parent graph. So, for instance, if we have a graph G and a subgraph H such that H has all vertices of G but lacks one edge, it is considered a proper subgraph of G.
Let's dive deeper. So, how would we formally define a proper subgraph?
I think it should include all the vertices and at least some edges, right?
Close! A proper subgraph H of G must be a subgraph where the vertex set of H is a proper subset of G's vertex set. Additionally, the edge set must consist of edges that are present in G. This means if G has vertices A, B, C, and D, for H to be a proper subgraph, it might only involve A and B along with the edge (A, B) but cannot include all edges connecting to D.
So if H has every vertex and edge that G has, then it’s not a proper subgraph?
Precisely! That is a key point to remember.
Now, let’s talk about induced subgraphs. Does anyone know what makes an induced subgraph unique?
It takes a subset of vertices and includes all edges between those vertices?
Exactly! Given a subset of vertices W from graph G, the induced subgraph contains all edges that connect vertices in W. For example, if W includes {A, C}, the induced subgraph will only include edges directly connecting A and C from G.
But could an induced subgraph also be a proper subgraph?
Yes, that’s correct. An induced subgraph can be a proper subgraph if it does not include all vertices or edges present in G.
Subgraphs are fun, subsets they make; proper ones differ, for clarity's sake.
Imagine a city (the graph) with various paths (edges). A subgraph is a journey through part of that city, while a proper subgraph must leave out at least one path.
Use 'SPIGG' (Subset, Proper, Induced Graph; Guide) to remember types of subgraphs.
PES: Proper Edges Subset - for remembering the characteristics of proper subgraphs.
Term
Subgraph
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Term
Induced Subgraph
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A graph formed from a subset of the vertices and edges of another graph.
A subgraph that contains at least one vertex or edge not present in the original graph.
A subgraph formed by a subset of vertices that includes all edges connecting those vertices.
The removal of an edge from a graph, which affects only the edge set.
The removal of a vertex from a graph, which affects both the vertex and its incident edges.
This section defines what constitutes a proper subgraph in graph theory, highlighting its characteristics and relation to induced subgraphs and various graph operations.
The section delves into the definition of a proper subgraph, contrasting it with regular subgraphs, and introduces the concept of induced subgraphs. It also covers various operations that can be performed on graphs, such as edge and vertex removal, and discusses the implications of these operations on the graph’s structure.
In graph theory, a proper subgraph is a specific type of subgraph that retains at least one element which is absent in its parent graph. This section breaks down the definition of a proper subgraph as one that is a subset of both the vertex and edge sets of the original graph, while also distinguishing it from induced subgraphs, which focus solely on specific vertices and their corresponding connections. The content further explores basic operations on graphs, particularly the effects of adding or removing edges and vertices on the structure of the graph. Understanding these concepts is crucial for more advanced topics in graph theory, such as graph isomorphism and connectivity.