An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand, I. Charon, O. Hudry and A.… (More)
The game chromatic number χg is considered for the Cartesian product G 2 H of two graphs G and H. We determine exact values of χ g (G2H) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χg(G2H) is not bounded from above by a function of game chromatic numbers of graphs G and H. An analogous result is… (More)
A profile on a graph G is any nonempty multiset whose elements are ver-tices from G. The corresponding remoteness function associates to each vertex x ∈ V (G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary… (More)
In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpi´nski graphs. We compute the minimum size of such codes in Sierpi´nski graphs.
The median (antimedian) set of a profile π = (u 1 ,. .. , u k) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness i d(x, u i). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles… (More)
The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r ≥ 2, there exists a connected graph H such that G is the… (More)
For a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exists (a, b)-codes in Sierpi´nski graphs.
Eppstein  introduced semicube graphs as the key tool for efficient computation of the lattice dimension of a graph. In this paper it is shown that, roughly speaking, every graph can be realized as the semicube graph of some partial cube. Terminal semicubes and terminal expansions are introduced along the way. Semicube graphs of trees are studied in… (More)