Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture

In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open.Comment: Final version. To appear in European Journal of Combinatoric

The obstructions for toroidal graphs with no K3,3K_{3,3}'s

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no $K_{3,3}$-subdivisions that coincide with the toroidal graphs with no $K_{3,3}$-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no $K_{3,3}$'s and prove that the lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail

Beyond the stereotypes: An exploration of female counsellors’ experience of working with male survivors of sexual abuse

This study examines the experience of female counsellors working with male survivors of sexual abuse in the context of the social conditions which are brought into therapy by both the counsellor and the client. The concept of male sexual abuse stands in stark contrast to the notion of masculinity and male survivors often face the additional distress of their perceived failure to live up to our cultural notions of what it is to be a real man. Counsellors are exposed to the same cultural conditioning and bring preconceptions into therapy. This qualitative phenomenological study explored the experience of four female person-centred counsellors using semi-structured face to face interviews. The findings indicate that the counsellors were aware of the impact of social conditioning on their clients and recognised the additional distress this may cause. They were also aware of their own preconceptions about masculinity but seemed less conscious of the lingering impact that traditional gender role stereotyping has on their emotional responses to their clients. The participants found this work challenging personally and yet their commitment to their clients enabled them to offer a deep therapeutic relationship. Their clients were ready to work and fully engaged in the therapeutic process and somehow together they managed to find a healing space beyond the stereotypes

Orientably regular maps with given exponent group

We prove that for every integer d ≥ 3 and every group U of units mod d, there exists an orientably-regular map of valency d with exponent group U

On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar