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- Title
The edge-to-edge geodetic domination number of a graph.
- Authors
John, J.; Flower, V. Sujin
- Abstract
Let G = (V, E) be a connected graph with at least three vertices. A set S Ç E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G} of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m -- 1 are characterized. We proved that if G is a connected graph of size m > 4 and G is also connected, then 4 < γee(G) +γsee(G) < 2m -- 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 < a < b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a < b, there exists a connected graph G with γ(G) = a and 'ygee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.
- Subjects
DOMINATING set; GEODESICS; GRAPH connectivity; INTEGERS
- Publication
Proyecciones - Journal of Mathematics, 2021, Vol 40, Issue 3, p635
- ISSN
0716-0917
- Publication type
Article
- DOI
10.22199/issn.0717-6279-4057