Various Operations on Graphs

Today, we will start by discussing subgraphs. Can anyone tell me what a subgraph is?

Is it a smaller graph made from some vertices and edges of a bigger graph?

Exactly! A subgraph is a graph formed from a subset of a graph's vertices and edges. For example, if we have a graph G with vertices V and edges E, then a subgraph H will have its vertices as a subset of V and edges as a subset of E. Remember, all vertices in H must belong to G!

What about proper subgraphs? Are they different?

Great question! A proper subgraph is indeed a type of subgraph, but with a key difference: it should not be equal to the original graph G. So it must have fewer vertices or edges.

Can you give us an example of a proper subgraph?

Sure! If G has vertices A, B, and C with edges AB and BC, a proper subgraph H could have vertices A, B and edge AB. However, H can't have all vertices and edges as G because it wouldn't be a proper subgraph.

Just to recap: A subgraph uses vertices and edges from G, and a proper subgraph specifically must not equal G. Any further questions?

Next, let's talk about induced subgraphs. Can anyone explain what they are?

Are they the same as normal subgraphs?

Good question! Induced subgraphs are a specific type of subgraph. For a given subset of vertices W, the induced subgraph includes all edges that connect any two vertices in W.

So, if W has no edges, the induced subgraph will have no edges too?

Exactly! If W is a single vertex, the induced subgraph is simply that vertex with no edges. An important aspect to remember: edges are only included if both endpoints are in W.

What if we take a subset that is empty?

Good point! An induced subgraph with an empty set of vertices is effectively an empty graph. Always remember the edges depend on the vertices selected!

In summary, the induced subgraph is formed from a specified subset of vertices, including all edges connecting them. Any more questions?

Now, let’s discuss operations on graphs, starting with edge deletion. Who can summarize what happens if we delete an edge?

The edge is removed, but the vertices stay the same, right?

Correct! Deleting an edge only modifies the edge set while keeping the vertex set intact. This is similar to how removing a cable from a computer network doesn't remove the computers themselves.

Got it! What about vertices?

Good question! When we delete a vertex, we also must remove any edges connected to that vertex, since those connections will no longer exist. Make sure to remember: deleting a vertex affects both the vertex and edge sets!

What if we add a new edge? Does that affect the vertex set too?

Yep! Adding a new edge might introduce new vertices into the graph as well! Always remember: operations can change both vertex and edge sets depending on what's being altered.

To recap: edge deletions keep vertex sets intact, while removing a vertex affects both sets. Adding edges can expand the vertex set. Any further clarifications?

Let’s shift gears to graph isomorphism. Can anyone explain what it means for two graphs to be isomorphic?

Is it like saying they are the same graph but just labeled differently?

Exactly! When we say two graphs are isomorphic, we mean there is a one-to-one correspondence between their vertices such that their edges match up. The structure is the key part!

So, if two graphs look different but can be relabeled to become the same, are they isomorphic?

That’s correct! As long as you can find a bijective mapping that preserves the connections, the graphs are considered isomorphic. Remember, though, that it is essential to keep edge correspondences in mind!

What happens if they have a different number of vertices?

Excellent point! If they have a different number of vertices, they cannot be isomorphic. So, analyzing structural properties like that is crucial.

To summarize: Isomorphism checks for structural similarity through vertex correspondences, and differing vertex counts mean they cannot be isomorphic. Any questions?