1. Point dominance set: that is, the vertices in V are either elements in the V set or are adjacent to a vertex in V.
Communication example: To establish communication base stations between several cities, the direct connection means that the signal can be covered, at least how many cities need to build base stations?
The definition of the point dominance set: the point dominance set V ∗: ∀ v ∈ (V − V ∗), ∃ v ∗ ∈ V ∗ (v ∗, v) ∈ E The definition of the point dominance set: the point dominance of the graph G Set V^*:\forall v\in (VV^*),\exists v^* \in V^* (v^*,v)\in E Point supported with a set of predetermined sense : Fig G the point of support with set V∗:∀v∈( V−V∗),∃v∗∈V∗ (v∗,v )∈E
2. Point cover set: Point "covers" the edge, (the point is directly connected to an edge) In layman's terms, the so-called point cover set V* means that at least one vertex of all edges in G belongs to V*.
3. Point independent set: any two fixed points in the subset of the point set are not directly adjacent
relationship
The maximal independent set of G is the minimal dominating set of G. The inverse proposition does not hold (that is, the minimal dominating set is not necessarily the maximal independent set).
An independent set is a maximally independent set, if and only if it is a dominant set
Suppose there are no isolated vertices in the undirected graph G(V, E), and the set of vertices V ⊆V, then V is the point cover of G, if and only if V–V* is the independent set of points of G.
The minimal dominance set may not be a maximally independent set: eg: Mi The minimal dominance set may not be a maximally independent set: eg: m A very small branch with a set of not necessarily a very large independent stand set : such as : rice
Solve
The solution of the minimal point dominating set: each vertex and all its adjacent vertices are added together to form a factor term, and all the factor terms are multiplied.
All adjacent vertices of each vertex are subjected to product operation and then sum operation with the vertex to form a factor term; all the factor terms are then multiplied and expanded into the form of the sum of products according to the logic operation law. In the result obtained after the calculation, each product term represents a minimal covering set, and the smallest one is the smallest covering set.
There is complementarity between the minimum point cover set and the maximum point independent set of the undirected connected graph G. After the minimum point cover set and the point cover number are obtained, the maximum point independent set and the point independent number can be obtained. This results in a maximal independent set.
+ Is or, multiplication is and, commutative, associative, and distributive are all the same as numbers, and there is, absorption vi + vi = vi; vivi = vi; vi + vivj = vi. + Is or, multiplication is and, commutative, associative, and distributive laws are the same as numbers, and there are, absorption laws vi + vi = vi; vivi = vi; vi + vivj = vi.+ Is or , by a and , cross exchange law , knot together law , sub- distribution law are the number of words a kind , and there is , absorption yield law V I+v i=v i ；v i v i=v i ；v i+v i v j=v i 。
Dividing line Minute cut line
Edge Covering Set: Edges cover points, several edges can "cover" all fixed points
Independent set of edges (matching): Any two edges have no common vertices.
For a given matching M, if the edge (u, v) ∈ M, then the vertices u and v are said to be matched by M. u and v are the covered points of M, and the vertices that are not associated with the edges in matching M are called uncovered points. For a given matching M, if the edge (u, v) ∈ M, then the vertices u and v are said to be matched by M. u, v are the covered points of M, and the vertices that are not associated with the edges in matching M are called uncovered points.For in a th a given a match with M ,If edge ( u ,v )∈M , the said top point u and v are M being matched with . u ,v is the cover point of M ,Not to match with M in the edge with clearance associated to a top point referred to not cover the point .
relationship
Suppose the number of vertices of the undirected graph G is n, and there are no isolated points in G.
(1) Let M be a maximum match of G. For each uncovered point v of M in G,
the set of edges formed by an edge associated with v is selected as N, then W = M∪N is the value in G Minimum edge coverage;
(2) Let W1 be the minimum edge coverage of G. If there is an adjacent edge in G, remove one of them. Set the removed edge set to
N1, then M1 = W1-N1 is one of G Maximum matching;
(3) G satisfies: the number of edge covers + the number of independent edges = the number of vertices
Matching issues:
The problem of finding the general maximum match
Staggered track (for a matching M, there are staggered edges that belong to M or not belong to M)
Augmentable track: For a given matching M, both ends are uncovered staggered tracks (especially, if there is only one edge between the two uncovered points, then this edge alone will also form an augmentable track Wide rail)
The meaning of augmentable track (that is, matching M is expanded) : For a certain augmentable track P, remove the original M in P, and add the edges in P that did not belong to M to the new match. After the change The match of is exactly one side more than the original match M.
Theorem : M is the maximum matching of G, if and only if G does not have an augmentable orbit about M.
Therefore, a feasible method to find the maximum matching is: given an initial matching M (if not given, then M = Ø), if the graph G has no uncovered points, there will definitely be no augmentable orbit, that is, M It is the maximum matching;
otherwise, for all the uncovered points vi in the graph G, search for the
augmentable orbits with vi as the endpoint through a certain method , so as to gradually expand M through the augmentable orbits. (In the process of expanding M, some uncovered points will gradually be covered by M)
How to find the maximum matching of a bipartite graph:
- Network flow solution
- Hungary algorithm
- Hopcroft-Karp algorithm