Bennet Manvel

Hamiltonian cycles in cubic 3-connected bipartite planar graphs

Abstract We show that 3-connected cubic bipartite planar graphs with fewer than 66 vertices are Hamiltonian.

Reconstruction of sequences

Abstract Every sequence of length n determines ( n k ) subsequences of length k . We investigate the relationship between such subsequences and the original sequence. In particular, we show that for n >;7 and k ⩾[ n /2] the subsequences uniquely determine the original sequence, and for k 2 n they do not.

Some basic observations on Kelly's conjecture for graphs

P.J. Kelly first mentioned the possibility of determining a graph from subgraphs obtained by deleting several points. While such problems have received a great deal of attention in the case of deletions of single points, the problem for several points is virtually untouched. This paper contains some basic results on that problem, including the negative observation that for every k, there exist two non-isomorphic graphs with the same collection of k-point subgraphs.

Nearly acyclic graphs are reconstructible

We prove that a graph G is reconstructible if G has a node v with G-v acyclic. The proof uses colored graphs and shows how to reconstruct some graphs from pieces which share a common subgraph having few automorphisms.

Reconstruction of maximal outerplanar graphs

S. Ulam has conjectured that every graph with three or more points is uniquely determined by its collection of point-deleted subgraphs. This has been proved for various classes of graphs, but progress has generally been confined to very symmetrical graphs and graphs with connectivity zero or one. Although the algorithms used in those cases do not generalize to graphs with higher connectivity, they do employ methods which might be applied more widely. In this paper, maximal outerplanar graphs are reconstructed by an algorithm which exploits their special structure. In fact it is shown that such a graph is determined by its set of non-isomorphic point-deleted subgraphs.

On reconstructing graphs from their sets of subgraphs

Abstract Using only the set of point-deleted subgraphs, several invariants of a graph are derived. That information is then used to show that disconnected graphs and separable graphs without endpoints are reconstructible from their sets of deleted subgraphs. Similar results are obtained in the line-deleted case.

Reconstructing the degree pair sequence of a digraph

Abstract It is shown that the degree pair sequence of any digraph with five or more points can be derived from its one point-deleted subdigraphs.

Separation of graphs into three components by the removal of edges

Several ways to separate a connected graph into three components by the removal of edges are discussed. Graphical parameters that count the number of edges removed are introduced and the relations between these parameters are given.

Determining connectedness from subdigraphs

Abstract A method is given by which the connectedness category of an arbitrary digraph D with more than four points can be determined from its subdigraphs D i = D − v i .