Graph structural properties of non-Yutsis graphs allowing fast recognition

Yutsis graphs are connected simple graphs which can be partitioned into two vertex-induced trees. Cubic Yutsis graphs were introduced by Jaeger as cubic dual Hamiltonian graphs, and these are our main focus. Cubic Yutsis graphs also appear in the context of the quantum theory of angular momenta, where they are used to generate summation formulae for general recoupling coefficients. Large Yutsis graphs are of interest for benchmarking algorithms which generate these formulae. In an earlier paper we showed that the decision problem of whether a given cubic graph is Yutsis is NP-complete. We also described a heuristic that was tested on graphs with up to 300,000 vertices and found Yutsis decompositions for all large Yutsis graphs very quickly. In contrast, no fast technique was known by which a significant fraction of bridgeless non-Yutsis cubic graphs could be shown to be non-Yutsis. One of the contributions of this article is to describe some structural impediments to Yutsisness and to provide experimental evidence that almost all non-Yutsis cubic graphs can be rapidly shown to be non-Yutsis by their application. Combined with the algorithm Preprint submitted to Discrete Applied Mathematics 5 October 2007 described in the earlier paper this gives an algorithm that, according to experimental evidence, runs efficiently on practically every large random cubic graph and can decide on whether the graph is Yutsis or not. The second contribution of this article is a set of construction techniques for nonYutsis graphs implying, for example, the existence of 3-connected non-Yutsis cubic graphs of arbitrary girth and with few non-trivial 3-cuts.