It is proved that the metric dimension of G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$.Expand

The first contribution is to characterize the graphs in Gβ,D with order β+D for all values of β and D and to determine the maximum order of a graph in the set of graphs with metric dimension β and diameter D.Expand

This is the first paper to investigate arch layouts, and characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k-1-colourable.Expand

It is proved that the queue-number is bounded by the tree-width, thus resolving an open problem due to Ganley and Heath and disproving a conjecture of Pemmaraju.Expand

The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O (log min\sn(G),qn(G) ), which reduces the question of whether queue-number is bounded by stack-number to whether 3- stack graphs have bounded queue number.Expand

It is proved that every graph with treewidth $k$ and maximum degree $\Delta$ has a $O(k\Delta)$-colouring that is nonrepetitive on paths, and a “O( k\Delta^3)”-colours that isNonre petitive on walks.Expand

This thesis introduces and comprehensively study so-called track layouts of graphs and their subdivisions, and establishes that graphs of bounded treewidth have three-dimensional straight-line grid drawings with linear volume.Expand

It is proved that every proper minor-closed class of graphs has bounded queue-number, and it is shown that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth.Expand