Question: In the context of the Max-Cut problem, I'm trying to understand the distribution of graphs based on their edge-cut properties. Specifically, I'm interested in whether there are generally more graphs where all edges can be cut (i.e., each edge connects vertices in different partitions), or if it's more common to find graphs where not all edges are cut.
Background: The Max-Cut problem is a fundamental problem in combinatorial optimization, where the objective is to partition the vertices of a graph into two disjoint subsets in such a way that the number of edges between the two subsets is maximized. This problem is NP-hard, indicating that it is challenging to find an optimal solution for large instances.
What I Know: I understand that the answer might depend on the types of graphs in question (e.g., simple, undirected graphs versus directed graphs, or sparse versus dense graphs). For simple, undirected graphs, it seems intuitive that having all edges cut might be less common, especially as the number of vertices increases, due to the combinatorial increase in possible edge configurations.
What I've Tried: I've done some preliminary research on the topic, looking into graph theory textbooks and combinatorial optimization literature, but most resources focus on algorithms for solving the Max-Cut problem rather than on the statistical distribution of graph types based on their edge-cut properties.
My Question: Can anyone provide insights or references that discuss the prevalence of graphs where all edges are cut versus those where not all edges are cut in the context of the Max-Cut problem? Are there known results or studies that explore this aspect of graph theory?
Additional Context: I'm particularly interested in any theoretical results or computational studies that might give a sense of how common each type of graph is, especially in applications where understanding the structure of potential solutions to the Max-Cut problem is important.
