4 research outputs found
Traversing Every Edge in Each Direction Once, But Not at Once: Cubic (Polyhedral) Graphs
A {\em retracting-free bidirectional circuit} in a graph is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Such a circuit revisits each vertex only in a number of steps. Studying the class of all graphs admitting at least one retracting-free bidirectional circuit was proposed by Ore (1951) and is by now of practical use to nanotechnology. The latter needs in various molecular polyhedra that are constructed from a single chain molecule in the retracting-free way. Some earlier results for simple graphs, obtained by Thomassen and, then, by other authors, are specially refined by us for a cubic graph . Most of such refinements depend only on the number of vertices of
Vertex Splitting and Upper Embeddable Graphs
The weak minor G of a graph G is the graph obtained from G by a sequence of
edge-contraction operations on G. A weak-minor-closed family of upper
embeddable graphs is a set G of upper embeddable graphs that for each graph G
in G, every weak minor of G is also in G. Up to now, there are few results
providing the necessary and sufficient conditions for characterizing upper
embeddability of graphs. In this paper, we studied the relation between the
vertex splitting operation and the upper embeddability of graphs; provided not
only a necessary and sufficient condition for characterizing upper
embeddability of graphs, but also a way to construct weak-minor-closed family
of upper embeddable graphs from the bouquet of circles; extended a result in J:
Graph Theory obtained by L. Nebesk{\P}y. In addition, the algorithm complex of
determining the upper embeddability of a graph can be reduced much by the
results obtained in this paper