Having the dog makes Anna more accessible, since she must take the dog out for walks. It also makes her easier to meet. Somerset Maughamwrote in one of his Ashenden stories that it is never hard to meet a person who owns a dog. Strangers start by showing an interest in the dog, petting it, asking questions about it, and so on. It is possible that Anna has brought the little dog along on her vacation trip because she is hoping, consciously or consciously, that the cute little animal will be helpful in meeting people. The dog plays no further part in the story after it has brought Anna and Gurov together. Having a pet dog also shows that Anna is lonely. Most people keep dogs, cats, and birds for company.
Chekhov states that Gurov has had many extramarital affairs in the past, but what makes his affair with Anna different is that they are both away from their homes and families and have an unusually long time to bond with each other. Gurov's previous affairs may have been much briefer and more furtive, and this is a first time for Anna. The central part of the story is essential because it takes a certain length of time for the pair to form a bond, and the bonding is the whole point of the story.
There seems to be four parts (though this is somewhat arbitrary): Yalta; Moscow; Anna's Town; Moscow Redux. Each section sees a slightly altered Gurov: in Yalta, he is the sexist (misogynist?) that looks at women as inferior, even treating Anna as a plaything that he will soon discard. Here, Gurov treats Anna with no more concern than he would, say, a lapdog.
Back in Moscow, Gurov begins to to think about Anna, seeing her as a person, perhaps, for the first time. He even gets upset when he begins to open up about Anna and his acquaintance can only comment about the stinky fish.
Gugrov becomes a bit more obsessed, and journeys to Anna's town, like a dog coming home seeking the warmth of its master. He tracks her down, and they begin their affair on a more regular basis.
Finally, returning to Moscow, Anna and Gurov continue their affair even in the midst of an uncertain future. Gurov is getting old, and he begins to think that his life has passed him by. Anna represents his chance to live again, or, more cynically, to recapture his youth.
This story is about the difficulties of connecting. It’s about relationships of love and the subtle power plays within those relationships.
It seems to me that the lapdog has been replaced by Gurov at the story’s end. (Of course this is not the only interpretation, but it is fun.)
The story can be divided into three parts a) meeting of Gurov and Anna at yalta where they get aquainted with each other for the first time b) the period when Anna and Gurov both living separetly and thinking of each other, at the same time both of them feel that they were happy when were together the third and the most important part c) when they meet up again and start thiking of a relationship which will go on in future...how to avoid maintaing of secrecy of their relationship,acknowledging their love etc.
Chekhov has kept a dog with the lady which is the sign of prestige and aristocracy.....because Anna is first introduced in story by 'lady with the dog' which indicates that she is from well to do family or class maintians high class life and so on.