Abstract
The distance d G ( u , v ) between two vertices u and v in a connected graph G is the length of the shortest ( u , v ) path in G . A ( u , v ) path of length d G ( u , v ) is called a ( u , v ) - geodesic. A set X ⊆ V is called weakly convex in G if for every two vertices a , b ∈ X , exists an ( a , b ) -geodesic, all of whose vertices belong to X . A set X is convex in G if for all a , b ∈ X all vertices from every ( a , b ) -geodesic belong to X . The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G , while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G . In this paper we consider weakly convex and convex domination numbers of tori.
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