The relationship between a master set, a vertex overwrite set, and a vertex independent set.

1. If no vertex is encouraged in an undirected graph G (u, v), then the extremely large vertex independent set of G is the extremely small Dominant Set of G. The inverse proposition is not true.

Understanding: If v * is set to a very big vertex independent set of G, then for those vertices that do not belong to V *, they must be connected to a vertex in V * (otherwise, it will not be great ), therefore, V * must be a dominant set. However, because all vertices in V * are independent, after a vertex is taken out of V *, the vertex cannot be connected to any remaining vertices in V, that is, it cannot be dominated. Therefore, in this condition, V * is the smallest Dominant Set.

2. An independent set is a huge independent set, if and only if it is a dominant set.

Understanding: If v * is set to a very big vertex independent set of G, then for those vertices that do not belong to V *, they must be connected to a vertex in V * (otherwise, it will not be great ), therefore, V * must be a dominant set.

3. No isolated point exists in an undirected graph G (V, E). If the vertex set V * is included in V, V * is the vertex overwrite of G, when and only when the V-V * is a point independent set of G.

Understanding: Set T as the V-V * (that is, V * on the V of the complementary set), then, if T is a point independent set, it indicates that no side of the two related points belong to T, therefore, at least one of the two vertices of all edges belongs to T, that is, V *. Then v * is the vertex overwrite set of G. In turn, if v * is the point coverage of G, it indicates that at least one of the two vertices of all edges belongs to V *. Therefore, in T, it is impossible to have two adjacent points, that is, T is the vertex independent set.

Inference: If G is a graph of order n without an isolated point, then v * is the extremely small (minimum) Point Coverage of G, when and only when the V-V * Is G's extremely large (maximum) vertices are independent sets, so that the number of vertices overwrites + the number of vertices independent = n.

4. minimum vertex overwrite = maximum matching of a two-part Graph

understanding: because we need to cover all the edges with as few points as possible, for any of the X and Y parts of the bipartite graph, if an edge is connected to a vertex as few as possible (for any part, it is better to connect to only one vertex), then this is the matching problem. However, we have to overwrite all the edges, so we need the maximum matching. When we obtain the maximum matching, all other edges can be overwritten by vertices with the largest match. (If an edge that cannot be overwritten exists, it is not the largest match ).