The edgetoedge geodetic domination number of a graph
DOI:
https://doi.org/10.22199/issn.071762794057Keywords:
Edgetoedge geodetic domination number, Edgetoedge geodetic number, Edge domination number, Geodetic numberAbstract
Let G = (V, E) be a connected graph with at least three vertices. A set S Í E is called an edgetoedge geodetic dominating set of G if S is both an edgetoedge geodetic set of G and an edge dominating set of G. The edgeto edge geodetic domination number ¡gee(G) of G is the minimum cardinality of its edgetoedge geodetic dominating sets and any edgetoedge geodetic dominating set of minimum cardinality is said to be a gee set of G. Some general properties satisfied by this concept are studied. Connected graphs of size m?2 with edgetogeodetic domination number 2 or m or m1 are charaterized. We proved that if G is a connected graph of size m ? 3 and G is also connected,then 4 ?¡gee(G) + ¡gee(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 and b with 2 ?a ? b, there exists a connected graph G with gee(G) = a and ¡gee(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 ¡e(G) = a and¡ gee(G) = b, where ¡e(G) is the edge domination number of G and gee(G) is the edgetoedge geodetic number of G.
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