Sylvain Gravier

Total domination number of grid graphs

We use the link between the existence of tilings in Manhattan metric with {1}-bowls and minimum total dominating sets of Cartesian products of paths and cycles. From the existence of such a tiling, we deduce the asymptotical values of the total domination numbers of these graphs and we deduce the total domination numbers of some Cartesian products of cycles. Finally, we investigate the problem of total domination numbers for some Cartesian products of two paths.

Identifying codes of cycles

In this paper we deal with identifying codes in cycles. We show that for all r ≥ 1, any r-identifying code of the cycle Cn has cardinality at least gcd(2r + 1, n) ⌈n/2gcd(2r+ 1,n)⌉. This lower bound is enough to solve the case n even (which was already solved in [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (7) (2004) 969-987]), but the case n odd seems to be more complicated. An upper bound is given for the case n odd, and some special cases are solved. Furthermore, we give some conditions on n and r to attain the lower bound.

Paired-domination in generalized claw-free graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206). A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by

Graphs whose choice number is equal to their chromatic number

\gamma_{\rm pr}(G)

Some results on total domination in direct products of graphs

, is the minimum cardinality of a paired-dominating set of G. If G does not contain a graph F as an induced subgraph, then G is said to be F-free. Haynes and Slater (Networks 32 (1998) 199–206) showed that if G is a connected graph of order

On domination numbers of Cartesian product of paths

n \ge 3