- Tytuł:
- Geodetic sets in graphs
- Autorzy:
-
Chartrand, Gary
Harary, Frank
Zhang, Ping - Powiązania:
- https://bibliotekanauki.pl/articles/743733.pdf
- Data publikacji:
- 2000
- Wydawca:
- Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
- Tematy:
-
geodetic set
geodetic number
upper geodetic number - Opis:
- For two vertices u and v of a graph G, the closed interval I[u,v] consists of u, v, and all vertices lying in some u-v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u,v] for u, v ∈ S. If I[S] = V(G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for every two distinct vertices u,v ∈ S, there exists a third vertex w of G that lies in some u-v geodesic but in no x-y geodesic for x, y ∈ S and {x,y} ≠ {u,v}. It is shown that for every integer k ≥ 2, there exists a connected graph G with g(G) = k which contains a uniform, essential minimum geodetic set. A minimal geodetic set S has no proper subset which is a geodetic set. The maximum cardinality of a minimal geodetic set is the upper geodetic number g⁺(G). It is shown that every two integers a and b with 2 ≤ a ≤ b are realizable as the geodetic and upper geodetic numbers, respectively, of some graph and when a < b the minimum order of such a graph is b+2.
- Źródło:
-
Discussiones Mathematicae Graph Theory; 2000, 20, 1; 129-138
2083-5892 - Pojawia się w:
- Discussiones Mathematicae Graph Theory
- Dostawca treści:
- Biblioteka Nauki
Artykuł