The signed domatic number of some regular graphs
Discrete Applied Mathematics
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The distanced"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.