Minors for alternating dimaps

We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.Comment: 51 pages, 7 figure

Initial experience in self-monitoring of intraocular pressure.

Background/aims: Diurnal variation in intraocular pressure (IOP) is a routine assessment in glaucoma management. Providing patients the opportunity to perform self-tonometry might empower them and free hospital resource. We previously demonstrated that 74% of patients can use the Icare® HOME tonometer. This study further explores Icare® HOME patient self-monitoring. Methods: Patients were trained by standard protocol to use the Icare® HOME rebound tonometer. Patient self-tonometry was compared to Goldmann applanation tonometry (GAT) over one clinical day. Following this, each patient was instructed to undertake further data collection that evening and over the subsequent two days. Results: Eighteen patients (35 eyes) participated. Good agreement was demonstrated between GAT and Icare® HOME for IOPs up to 15 mm Hg. Above this IOP the Icare® tended to over-read, largely explained by 2 patients with corneal thickness >600 um. The mean peak IOP during ‘clinic hours’ phasing was 16.7 mm Hg and 18.5 mm Hg (p = 0.24) over three days. An average range of 5.0, 7.0 and 9.8 mm Hg was shown during single day clinic, single day home and three day home phasing respectively (p =<0.001). The range of IOP was lower in eyes with prior trabeculectomy (6.1 mm Hg vs 12.2 mm Hg). All patients undertook one reading in the early morning at home with an average of 4.8 readings during, and 3.1 readings after office hours. Conclusions: This small study shows that self-tonometry is feasible. The findings from home phasing demonstrated higher peak and trough IOPs, providing additional clinical information. Home phasing is a viable alternative. The cost-effectiveness of this approach has yet to be addressed