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Erdös has defined g(n) as the smallest integer such that any set of g(n) points in the plane, no three collinear, contains the vertex set of a convex n-gon whose interior contains no point of this set. Arbitrarily large sets containing no empty convex 7-gon are constructed, showing that g(n) does not exist for n≥l. Whether g(6) exists is unknown.

Let G (V, E) be a finite undirected graph with n vertices and m edges. A minimum edge dominating set of G is a set of edges D, of smallest cardinality 3"(G), such that each edge of E D is adjacent to someThe edge domination problem is NP-complete for planar bipartite graphs, their subdivision, line, and total graphs, perfect claw-free graphs, and planar cubic graphs. The stable set problem and the edge domination problem are NP-complete for iterated total graphs.The edge domination problem is solvable in O(n 3) time for claw-free chordal graphs, locally connected claw-free graphs, the line graphs of total graphs, the line graphs of chordal graphs, the line graph of any graph in which each nonbridge edge is in a triangle, and the total graphs of any of the preceding graphs.

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Abstract. An algorithm is given to solve the minimum cycle basis problem for regular matroids. The result is based upon Seymour's decomposition theorem for regular matroids; the Gomory-Hu tree, which is essentially the solution for cographic matroids; and the corresponding result for graphs. The complexity of the algorithm is O ((n + m) 4 ), provided that a regular matroid is represented as a binary n×m matrix. The complexity decreases to O ((n+m) 3.376 ) using fast matrix multiplication.