A graph-theoretic approach is taken to the component order problem in the layout of statistical graphics. Eulerian tours and Hamiltonian decompositions of complete graphs are used to ameliorate order effects in statistical graphics. Similarly, traversals of edge weighted graphs are used to amplify the visual effect of selected salient features in the data. Relevant graph theory is summarized and classic algorithms are tailored to this problem. Graphics for multiple comparisons are reviewed and a new display developed that is based on graph traversal. Improved star glyph displays of multivariate data are described. Parallel coordinate displays tailored to particular features of the data are developed. The methods and new graphical displays are made available as an R package, PairViz.