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By Herbert S. Wilf

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The remaining vertices of S are an independent set in a smaller graph, namely the graph that is obtained from G by deleting vertex v∗ as well as all vertices that are connected to vertex v∗ by an edge. This latter set of vertices is called the neighborhood of vertex v∗ , and is written N bhd(v∗ ). The set S consists, therefore, of vertex v∗ together with an independent set of vertices from the graph G − {v∗ } − N bhd(v∗ ). Now consider an independent set S that doesn’t contain vertex v∗ . In that case the set S is simply an independent set in the smaller graph G − {v∗ }.

7. Analyze the complexity of your algorithm maxset3 from exercise 6 above. 8. 4) to prove by induction that P (K; G) is a polynomial in K of degree |V (G)|. Then show that if G is a tree then P (K; G) = K(K − 1)|V (G)|−1. 9. Write out an algorithm that will change the vertex adjacency matrix of a graph G to the vertex adjacency matrix of the graph G/{e}, where e is a given edge of G. 10.

Hence suppose the theorem is true for all such multigraphs of fewer than m edges, and let G have m edges. We will construct an Eulerian circuit of G. Begin at some vertex v and walk along some edge to a vertex w. Generically, having arrived at a vertex u, depart from u along an edge that hasn’t been used yet, arriving at a new vertex, etc. The process halts when we arrive for the first time at a vertex v such that all edges incident with v have previously been walked on, so there is no exit. , we’re back where we started.

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Algorithms and Complexity by Herbert S. Wilf


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