Area-efficient static and incremental graph drawings

Abstract

In this paper, we present algorithms to produce orthogonal drawings of arbitrary graphs. As opposed to most known algorithms, we do not restrict ourselves to graphs with maximum degree 4. The best previous result gave an $(m - 1) \times \left( {\tfrac{m}{2} + 1} \right)$-grid for graphs with n nodes and m edges.

We present algorithms for two scenarios. In the static scenario, the graph is given completely in advance. We produce a drawing on a grid of size at most $\frac{{m + n}}{2} \times \frac{{m + n}}{2}$, or on a larger grid where the aspect ratio of the nodes is bounded. Furthermore, we give upper and lower bounds for drawings of the complete graph K _n in the underlying model. In the incremental scenario, the graph is given one node at a time, and the placement of previous nodes can not be changed for later nodes. We then come close to the bounds achieved in the static case and get at most an $\left( {\tfrac{m}{2} + n} \right) \times \left( {\tfrac{2}{3}m + n} \right)$-grid. In both algorithms, every edge gets at most one bend, thus, the total number of bends is at most m.

Then we focus on planar graphs and outer-planar graphs. We obtain planar drawings in an (m−n+1) x min $\left\{ {\tfrac{m}{2},m - n + 1} \right\}$-grid with m-n bends for planar triconnected graphs. The best previous result here was an m x m-grid and m bends, if the boxes of the nodes are constrained to be small.

All algorithms work in linear time.

The research was partly funded by the NIST Advanced Technology Program Award No. 70NANB5111162" and by the German Research Society, Grant DFG Ka/4-2.