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The external stability number of the graph

Lecture



The set TMX is called externally stable in the graph ( X, Γ ) if for each vertex of xpt there exists at least one image in the set T. The number b (G) = The external stability number of the graph called the number of external stability.

Examples: how many queens can be put (fig.12) or other figures, so that all the board fields are under attack (5 queens, 8 rooks or elephants, 12 horses), how many observers to put at the intersections of streets, so that all crossroads are supervised.

There is a fairly simple algorithm for finding the minimum externally stable set. For the graph G = (X, D) (see Fig. 13a), we construct the graph ( X, The external stability number of the graph , D ), where the mapping D of the set X to The external stability number of the graph so that The external stability number of the graph if y = x or y is a prototype of x (yql -1 x) (fig. 13b).

Next steps:

  1. remove from ( x, The external stability number of the graph , D ) vertices of x such that Dx M Dу for y№x (here we remove c, d, f ) and edges outgoing from them;
  2. if there is a "hanging edge" ( x, The external stability number of the graph ), include x in the sought-for set T and delete this vertex and its images (here a and FROM a and D are excluded from the graph, and Dа = { The external stability number of the graph };
    The external stability number of the graph
    Fig. 13
  3. repeat the preceding paragraphs until the impossibility of further reduction is found;
  4. one of the remaining vertices (for example, b ) is included in the set T and we exclude its images Db = { The external stability number of the graph };
  5. repeat the preceding paragraphs and get either T = ( a, b, e ) or T = ( a, b, g ).

Let be The external stability number of the graph - some graph. Subset of vertices The external stability number of the graph called a set of external stability if
one) The external stability number of the graph
2) The external stability number of the graph

The power of the minimal set of external stability is called the number of external stability of the graph. The external stability number of the graph . In order to find all the sets of external stability of the graph, it is necessary to find all the covers of the modified graph adjacency matrix. The modification is to add a single main diagonal.

Example

The external stability number of the graph
The external stability number of the graph The external stability number of the graph The external stability number of the graph The external stability number of the graph The external stability number of the graph The external stability number of the graph
The external stability number of the graph
The external stability number of the graph
The external stability number of the graph
The external stability number of the graph
The external stability number of the graph
The external stability number of the graph

External stability sets:

The external stability number of the graph

The external stability number of the graph
The external stability number of the graph

See also


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Discrete Math. Set theory. Graph theory. Combinatorics.

Terms: Discrete Math. Set theory. Graph theory. Combinatorics.