Planar Bus Graphs

Abstract

Bus graphs are used for the visualization of hypergraphs, for example in VLSI design. Formally, they are specified by bipartite graphs $$G=(B \cup V,E)$$ . The bus vertices B are realized by horizontal and vertical segments, and the connector vertices V are realized by points and connected orthogonally to the bus segments without any bend; this is called bus realization. The decision whether a bipartite graph admits a bus realization, where connections may cross, is NP-complete. In this paper we show that in contrast the question whether a planar bipartite graph admits a planar bus realization can be answered in polynomial time. First we deal with plane instances, i.e., with the case where a planar embedding is prescribed. We identify three necessary conditions on the partition $$B=B_{\mathrm {V}}\cup B_{\mathrm {H}}$$ of the bus vertices, here $$B_{\mathrm {V}}$$ denotes the vertical and $$B_{\mathrm {H}}$$ the horizontal buses. We provide a test whether a good partition, i.e., a partition obeying these conditions, exists. The test is based on the computation of a maximum matching on some auxiliary graph. Given a good partition we can construct a non-crossing realization of the bus graph on an $$O(n)\times O(n)$$ grid in linear time. In the second part we use SPQR-trees to solve the problem for general planar bipartite graphs.