Dominating sets, covering sets, independent sets and matching

Note: The following figure g are all undirected connected graphs

One, point domination

Disposable

For a point A in the vertex set V in Figure G has an edge link to another point B, called Point a dictates B.

"Point domination Set"

For a subset of the vertices in the vertex set V in Figure G V ', you can dictate the other points in V-v ', the point set V ' is the point domination set.

"Minimum Domination set"

For the dominating set V, any true subset of him is not a dominating set, which is called V is the minimum dominating set.

"Minimum Domination set"

The minimum dominating set of top points is the minimum dominating set.

Note: The minimum dominating set and the minimum dominating set are not a meaning.

For example, there is a minimum dominating set {1,3,5} in the current diagram because any of his true subsets are not dominating sets. But the minimum dominating set is {2,4}

"Point domination number"

The number of minimum point domination points is called the point domination number.

Point Domination Set properties: when a graph does not have an orphaned vertex, if V ' is a point dominating set, then V-v ' is also the point dominating set.

Second, point coverage

Coverage

For a point on figure G, a is known to have one side e linked to it, and point a covers the side E. (Here is the point for the edge of the overlay)

"Point Overlay Set"

For figure G, the subset V of its vertex set V ' can overwrite all edges in its edge set E. Then V ' is the point cover set.

"Minimum coverage set"

If any true subset of the point overlay set V ' is not a point overlay set, then the V ' is a minimum coverage set.

"Minimum point overlay set"

The point cover set with the least number of vertices is called the minimum point cover set.

Note: Minimum and minimum point cover sets are not the same concept, with the same minimum point dominating set and the smallest point dominating set.

"Point Overlay Number"

The number of vertices in the minimum point coverage set is called the number of point overlays.

Third, point Independent

Independent

In Figure g, if point A is directly connected to point B without an edge, the two points are not adjacent to each other. This is called the two-point independence. (This is about the independence of the point)

"Point Independent set"

In Figure G, any two points in the subset V ' of the vertex set V ' are independent of each other, then V ' is a point independent set.

"Maximum point independent set"

If any other vertex is added to the vertex subset V ', then V ' will not be a point independent set, which is called V ' is a maximal point independent set.

"Maximum point independent set"

The point independent set with the highest number of vertices is the maximum point independent set.

"Point independent number"

The number of vertices in the maximum point independent set is called the Point independent number.

Four, side coverage

Coverage

In Figure G, for an edge E with his end point A, B, we say that e covers points a, B. (Here is the overlay on the side)

"Edge Overlay Set"

In Figure G, for the subset E ' of the set E ' of the edge, all the edges of which can be overridden by all points in the vertex set V in Figure G, then E ' is the edge overlay set.

"Minimal Edge Overlay Set"

If any true subset of the edge overlay set E ' is not an edge overlay set, then the E ' is a minimal edge overlay set.

"Minimum Edge Overlay Set"

The edge overlay set with the fewest number of edges is called the minimum edge overlay set.

"Edge Overlay Number"

The number of edges in the minimum edge overlay set is called the edge overlay number.

Five, side independent (matching)

Independent

In Figure G, if Edge A and edge B do not have a common vertex, the two edges are said to be independent of each other. (This is the side of independence)

Edge Independent set (match)

In Figure G, for the subset E ' of the set E ' of the edge, any of the two edges are independent, and set E ' is called the Edge Independent set. Also known as E ' for the match of Figure G

"Maximal match (max Edge Independent Set)"

In Figure G, for the subset E ' of an edge's set E ', if E ' is an edge-independent set and then any one edge is no longer an edge-independent set, then E ' is called a great match (maximal edge Independent set).

Maximum match (maximum edge Independent set)

The match with the most number of edges is called the maximum match (maximum Edge Independent set).

"Edge Independent Number"

The number of edges in the maximum match is called the Edge Independent number

"Cover point and no cover point"

For a given match m for figure G (U,V):

1, if a point v in the vertex set V is the end point of an edge in M, then V is the cap of M.

2, If a point v in the vertex set V is not the end point of any of the sides in M, then the "V" is the non-cap of M.

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Dominating sets, covering sets, independent sets and matching