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arXiv · 1403.0342

A Generalisation of Isomorphisms with Applications

Josef Lauri·Russell Mizzi·Raffaele Scapellato
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Abstract

In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms. TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems. In particular, we mention the Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and Stability of Graphs. We start with a study of invariance under TF-isomorphisms. In particular, we show that alternating trails and incidence double covers are conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms between graphs or digraphs. We then define an equivalence relation and subsequently relate its equivalence classes to the incidence double cover of a graph. By directing the edges of an incidence double cover from one colour class to the other and discarding isolated vertices we obtain an invariant under TF-isomorphisms which gathers a number of invariants. This can be used to study TF-orbitals, an analogous generalisation of the orbitals of a permutation group.

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Josef Lauri, Russell Mizzi, Raffaele Scapellato. 2014-03-03. A Generalisation of Isomorphisms with Applications. https://arxiv.org/abs/1403.0342

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