Factor-critical graph
A factor-critical graph is like a group of friends playing a game of cards. Imagine you have a group of friends who love playing cards. They decided to play with a bunch of cards but they didn't have the same number of cards for everyone. Some had more, some had less. So they had to figure out how to make it fair for everyone.
In the same way, a factor-critical graph is a group of connected points (also called vertices) that are joined by lines (also called edges) such that removing any one of these lines would result in a graph that has fewer factors than the original graph. Factors in this context refer to independent sets of vertices which means they are a group of vertices that are not connected by any edges. So, removing any edge would mean you would lose an independent set of vertices.
To better understand this concept let's imagine we have a graph of points and lines. If we remove any line (edge), we would have a smaller graph. However, not all graphs are created equal. Some graphs have special properties, like the factor-critical graph where a removal of any edge would lead to a loss of independent sets of vertices.
So imagine that you have a necklace made of beads, where each bead represents a vertex, and the string connecting the beads represents the edge. Now, if you were to remove any string, some of the beads would fall off, meaning that some vertices would no longer be connected. Not all necklaces are factor-critical, only those where a removal of any string lead to the destruction of independent sets of vertices.
In summary, think of a factor-critical graph as a group of points connected with lines or edges, and if you remove any of the edges you would lose independent sets of vertices. This is similar to how if you remove a string from a necklace, you would lose some beads.
In the same way, a factor-critical graph is a group of connected points (also called vertices) that are joined by lines (also called edges) such that removing any one of these lines would result in a graph that has fewer factors than the original graph. Factors in this context refer to independent sets of vertices which means they are a group of vertices that are not connected by any edges. So, removing any edge would mean you would lose an independent set of vertices.
To better understand this concept let's imagine we have a graph of points and lines. If we remove any line (edge), we would have a smaller graph. However, not all graphs are created equal. Some graphs have special properties, like the factor-critical graph where a removal of any edge would lead to a loss of independent sets of vertices.
So imagine that you have a necklace made of beads, where each bead represents a vertex, and the string connecting the beads represents the edge. Now, if you were to remove any string, some of the beads would fall off, meaning that some vertices would no longer be connected. Not all necklaces are factor-critical, only those where a removal of any string lead to the destruction of independent sets of vertices.
In summary, think of a factor-critical graph as a group of points connected with lines or edges, and if you remove any of the edges you would lose independent sets of vertices. This is similar to how if you remove a string from a necklace, you would lose some beads.