National Conditions of Travel: http://www.nationalrail.co.uk/Conditions%20of%20Travel%202016.pdf
24.1 You may take up to two dogs or other small domestic animals free of charge with you unless a Train Company has set out any special conditions relating to their own train services. In such cases these conditions will be made available when buying your Ticket in advance, and will be shown on the Train Company’s website.
South Eastern Railway - "Your rights when travelling with us"https://www.southeasternrailway.co.uk/about-us/our-policies
You can travel on Southeastern services with a maximum of two small dogs on a lead or other small domestic animals free of charge. For each additional animal, a fee of half the adult fare of your ticket will be charged - up to a maximum of £5 for a single, and £10 for a return.
Pets other than dogs must be carried in a fully enclosed basket or pet carrier with dimensions not exceeding 85 x 60 x 60 cm and cannot be taken out of their basket or carrier. If your pet causes a nuisance, inconvenience or a safety hazard to other passengers you may be politely asked to remove your pet from the train or station property.
Dog breeds that are considered potentially aggressive, such as Greyhound, must wear a muzzle at all times during your journey and at stations.
You can travel nationally with up to two dogs ... but only two SMALL dogs on South Eastern? And who is their expert who has laid down that the greyhound (of all breeds!) is "potentially aggressive". I'm not in favour of BSL (Breed Specific Legislation) in general, and am open mouthed in amazement at the greyhound - of all breeds - being singled out.