In this paper we prove that given a finite collection of finite graphs, and the subsets of vertices of a random graph G that induce those graphs, it is almost always possible to uniquely reconstruct a class of graphs equivalent to G.

A definition is adopted for convexity of a set of directed lines in the plane. Following this, the duals of a number of standard problems of geometric probability are formulated. Problems considered in detail are the duals of Sylvester's problem, chord length distributions and Ambartzumian's combinatorial geometry. The paper suggests some questions for further work.

It is shown that an oriented graph of order n whose every indegree and outdegree is at least cn is hamiltonian if c ≥ ½ − 2−15 but need not be if c < ⅜.

We derive an explicit formula for the difference χ(G;λ) − χ(G|X;λ)χ(Tx(G); λ)/(λ − 1), where χ(G;λ) is the characteristic polynomial of a simple matroid G, G|X is the restriction of G to a flat X in G, and Tx(G) is the complete principal truncation of G at the flat X. Two counting proofs of this formula are given. The first uses the critical problem and the second uses the broken-circuit complex. We also derive several inequalities involving Whitney numbers of the first kind and other numerical invariants.

In a set of even cardinality n, each member ranks all the others in order of preference. A stable matching is a partition of the set into n/2 pairs, with the property that no two unpaired members both prefer each other to their partners under matching. It is known that for some problem instances no stable matching exists. In 1985, Irving found an O(n2) two-phase algorithm that would determine, for any instance, whether a stable matching exists, and if so, would find such a matching. Recently, Tan proved that Irving's algorithm, with a modified second phase, always finds a stable cyclic partition of the members set, which is a stable matching when each cycle has length two. In this paper we study a likely behavior of the algorithm under the assumption that an instance of the ranking system is chosen uniformly at random. We prove that the likely number of basic steps, i.e. the individual proposals in the first phase and the rotation eliminations, involving subsets of members in the second phase, is O(n log n), and that the likely size of a rotation is O((n log n)1/2). We establish a ‘hyperbola law’ analogous to our past result on stable marriages. It states that at every step of the second phase, the product of the rank of proposers and the rank of proposal holders is asymptotic, in probability, to n3. We show that every stable cyclic partition is likely to be almost a stable matching, in the sense that at most O((n log n)1/2) members can be involved in the cycles of length three or more.

We consider networks in which both the nodes and the links may fail. We represent the network by an undirected graph G. Vertices of the graph fail with probability p and edges of the graph fail with probability q, where all failures are assumed independent. We shall be concerned with minimising the probability P(G) that G is disconnected for graphs with given numbers of vertices and edges. We show how to construct these optimal graphs in many cases when p and q are ‘small’.