# Cauchy-Binet by Jim Morrow

By Jim Morrow

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Let S,, v E V*,be the set of edges of G* incident to vertex v , then S, is a cutset of G* since G* is 2-connected. Any cutset S of a graph G* is a sum of S, over all vertices v in a component of G* - S. Thus the cutset space of G* is generated by all these S,. Let u be an arbitrary vertex of G*, then S,, is a sum of S, over all v in VY - u. Thus B* = { S , : v E V* - u } is a basis of the cutset space of G*. Obviously every edge of G* is contained in at most two S, of B*. Thus the collection of cycles in G corresponding to cutsets of B* is a 2-basis of G.

I \ c - _ - - Fig. 14. A plane graph G and its geometric dual G*. Planar graphs: Theory and algorithms 16 Clearly the dual of the dual of a plane graph G is the original graph G. However a planar graph may give rise to two or more geometric duals since the plane embedding is not necessarily unique. 1 the plane embedding of G is essentially unique and hence the dual is unique. The following observation is often useful in designing an efficient algorithm for planar graphs. 4. Let G be a planar graph and G* be a geometric dual of G , then a set of edges in G forms a cycle (or cutset) in G ifand only ifthe corresponding set of edges of G*forms a cutset (res.

For example { v I , v2, v3, v4} is a minimum vertex cover of G in Fig. 1 (b). Every vertex cover must contain either of the two ends of each edge in a matching. Therefore the cardinality of the minimum vertex cover of G is no less than ( M ( G )1 . 8. 5. I f a planar connected graph G has minimum degree 3 or more, then the cardinality of a minimum vertex cover is at least min(jn/2], [ ( n 2)/31}. 1. What is an algorithm? Consider a computational problem on graphs, such as the planarity testing problem: given a graph, is it planar?

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