Non-Separable and Planar Graphs by Whitney H.

By Whitney H.

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9(G) , is the minimum number of edge-disjoint planar subgraphs whose union is G. A VLSI-designer has to place the cells on a printed circuit board (which usually consists of superimposed layers), according to several designing requirements. One of these requirements is to avoids crossings, since crossings lead to undesirable signals. It is therefore desired to find ways to handle wire crossings in the network. In practice, crossing wires must be laid out in different layers. There are two approaches for distributing the network to the layers.

G) + o(G) - 2, where ~(G) is the minimum edge-degree of G. To state the results in this subsection, we would like to introduce some notation. 1. LINE GRAPHICAL METHOD Let B C V(L) = E(G) and B = V(L) \ B such that EL(B) is a AL-cut of L. For any x E V(G), let = EL(B, B) = IEa(x) n BI, and p(x) = IEa(x) n BI. da(x) = p(x) + p(x). Let X = {x E V(G): p(x) > 0 and p(x) > O}. p(x) It is clear that Then AL = IEL(B) I = L p(x)p(x). xEX Now, we state a key result in this subsection due to Zamfirescu [272], from which several results on the edge-connectivity of line graphs of undirected graphs can be derived easily.

For example, in the layout of printed circuits one is interested in knowing if a graph corresponding a particular electronic network is planar, where the vertices correspond to electronic cells and the edges correspond to the conductor wires connecting the cells. Several linear algorithms for solving this problem have been proposed, see, for example, Even and Tarjn [94], and Liu [193]. If G is not planar, then, whenever it is drawn on the plane, some of its edges must cross. This rather simple observation gives rise to two basic concepts: crossing number and thickness.

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