A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

A set of vertices $S$ resolves a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let ${\cal G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in ${\cal G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in ${\cal G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in ${\cal G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known.

Given a graph G and a subset W ⊆ V (G), a Steiner W -tree is a tree of minimum order that contains all of W. Let S(W ) denote the set of all vertices in G that lie on some Steiner W-tree; we call S(W ) the Steiner interval of W. If S(W ) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G.Giventhe set of all vertices in G lying on some u-v geodesic, and let J [u, v] denote the set of all vertices in G lying on some induced u-v path. Given a set S ⊆ V (G), let I [S] = u,v∈S I [u, v], and let J [S] = u,v∈S J [u, v]. We call I [S] the geodetic closure of S and J [S] the monophonic closure of S. If I [S] = V (G), then S is called a geodetic set of G. If J [S] = V (G), then S is called a monophonic set of G. The minimum order of a geodetic set in G is named the geodetic number of G.In this paper, we explore the relationships both between Steiner sets and geodetic sets and between Steiner sets and monophonic sets. 140 C. Hernando et al. / Discrete Mathematics 293 (2005) 139 -154 and the geodetic number, and address the following questions: in a graph G when must every Steiner set also be geodetic and when must every Steiner set also be monophonic. In particular, amongothers we show that every Steiner set in a connected graph G must also be monophonic, and that every Steiner set in a connected interval graph H must be geodetic.

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