(graph theory) Amount of connectivity yielding solutions?

[Rainbow_Dash]

I have what I presume to be a basic knowledge of the terminology involved in graph theory which I will attempt to utilize in order to describe the problem in my mind.

Suppose an amount of nodes corresponding to a square number is arranged accordingly, forming n rows and n columns. Each node is to be assigned a value of either 1 or 0. An edge or connection is placed between two nodes if they are “orthogonally” adjacent in the arrangement, and if they both have a value of 1.
Given any (n x n) arrangement of the above description, how many of the 2^(n*n) possible combinations of 0s and 1s will result in connectivity between two specific opposing corner nodes?

An example of a connectivity yielding combination, or solution, for a 5 x 5 arrangement would be:

- - - - - - x
1 1 1 0 1
1 0 1 0 1
1 0 1 1 1
1 0 0 0 0
1 0 0 0 0
x

For all intents and purposes we will assert that a solution depends on the connectivity specifically between the lower-right and upper-left corner nodes, and no other two nodes. X is here to indicate the location of these two nodes; they are NOT nodes themselves.

Another distinct solution would be:

1 1 1 1 1
1 0 1 0 1
1 0 1 1 1
1 0 0 0 0
1 0 0 1 1

Keep in mind that despite being traversable by the same path as in the example above, in addition to numerous others, we are only considering connectivity, meaning that it has one, *or more*, traversable paths between the two specified nodes.
Finding an answer for very small arrangements, is quite simple.
Here are all the solutions for 2 x 2:

1 1
1 0

0 1
1 1

1 1
1 1

So 3 out of the 16 possible combinations are solutions that result in specified connectivity.
The answer for any arrangement could very well be determined using some computer program to iterate through all the possible combinations, finding which ones yield connectivity, but I would like to determine a mathematical/logical approach. So how would you go about this problem?

 

[mmwave]

Thank you for your question regarding the connectivity between two specific corner nodes in a graph theory problem. As a scientist with knowledge in this field, I would like to offer some insights and suggestions on how to approach this problem.

First, let's define some key terms to ensure we are on the same page. A graph is a mathematical structure consisting of nodes (also known as vertices) and edges (also known as connections). In your problem, the nodes represent the squares in the arrangement, and the edges represent the connections between them. The connectivity between two nodes refers to the existence of a path between them.

To solve this problem, we need to consider the properties of the graph. In this case, the graph is a grid with n rows and n columns, where each node can only have a value of 0 or 1. We also know that an edge is placed between two nodes if they are orthogonally adjacent (i.e. sharing a side) and both have a value of 1.

One approach to finding the number of possible combinations that result in connectivity between two specific corner nodes is to use a mathematical formula. We can calculate the total number of possible combinations by taking 2^(n*n), as each node can have two possible values (0 or 1). However, not all of these combinations will result in connectivity between the two specific corner nodes.

To narrow down the number of combinations, we can consider the properties of the corner nodes. In your example, the upper-left and lower-right corner nodes are always fixed at 1, and the remaining nodes can vary. This means that for any given arrangement, the number of combinations that result in connectivity between these two corner nodes will be the same. Therefore, we can focus on finding the number of combinations that result in connectivity between any two fixed corner nodes.

Next, we can consider the paths that connect the two corner nodes. In your example, you have shown two different solutions that result in connectivity between the two specified nodes. However, there can be multiple paths between these two nodes, as long as they are connected in some way. This means that for any given arrangement, there can be more than one combination that results in connectivity between the two corner nodes.

To find the number of possible combinations, we can use the concept of paths and combinations. Each path between the two corner nodes can be represented by a sequence of 0s and 1s. For example, in your