Let γ ( D ) denote the domination number of a digraph D and let C m □ C n denote the Cartesian product of C m and C n , the directed cycles of length n ≥ m ≥ 3 . Liu et al. obtained the exact values of γ ( C m □ C n ) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ ( C m □ C n ) for m = 6 , 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ ( C m □ C n ) for m = 3 k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C m □ C n with m up to 21. Moreover, the exact values of γ ( C n □ C n ) with n up to 31 are determined.

Toll convexity is a variation of the so-called interval convexity. A tolled walk T between u and v in G is a walk of the form T : u, w 1 , . . . , w k , v, where k ≥ 1, in which w 1 is the only neighbor of u in T and w k is the only neighbor of v in T . As in geodesic or monophonic convexity, toll interval betweenx lies on a tolled walk between u and v}. A set of vertices S is toll convex, ifFirst part of the paper reinvestigates the characterization of convex sets in the Cartesian product of graphs. Toll number and toll hull number of the Cartesian product of two arbitrary graphs is proven to be 2. The second part deals with the lexicographic product of graphs. It is shown that if H is not isomorphic to a complete graph, tn(G • H) ≤ 3 · tn(G). We give some necessary and sufficient conditions for tn(G • H) = 3 · tn(G). Moreover, if G has at least two extreme vertices, a complete characterization is given. Also graphs with tn(G • H) = 2 are characterized -this is the case iff G has an universal vertex and tn(H) = 2. Finally, the formula for tn(G • H) is given -it is described in terms of the so-called toll-dominating triples.

A tolled walk T between two non-adjacent vertices u and v in a graph G is a walk, in which u is adjacent only to the second vertex of T and v is adjacent only to the second-to-last vertex ofis the union of toll intervals between all pairs of vertices from S. The size of a smallest set S whose toll closure is the whole vertex set is called a toll number of a graph G, tn(G). This paper investigates the toll number of the strong product of graphs. First, a description of toll intervals between two vertices in the strong product graphs is given. Using this result we characterize graphs with tn(G ⊠ H) = 2 and graphs with tn(G ⊠ H) = 3, which are the only two possibilities. As an addition, for the t-hull number of G ⊠ H we show that th(G ⊠ H) = 2 for any non complete graphs G and H. As extreme vertices play an important role in different convexity types, we show that no vertex of the strong product graph of two non complete graphs is an extreme vertex with respect to the toll convexity.