In Topdog/Underdog, what is the central conflict between the characters? How does the writer describe the act of writing this play?

The central conflict between these characters revolves around who ultimately deserves the position of top-dog in the relationship. The writer describes the act of writing the play as an open-ended exploration into the human psyche; she maintains that Topdog/Underdog can be interpreted in a number of ways, and she revels in the beauty of writing such a play. Parks received inspiration for her play from another one of her works: The America Play.
In The America Play, the protagonist is also a Lincoln impersonator. In Topdog/Underdog, Parks uses another Lincoln impersonator to explore the intricacies and dynamics of a dysfunctional sibling relationship. In this, Parks lays bare her naturalist approach to drama. She allows her characters to act and to emote naturally; the expelling of raw emotion is both an authentic and cathartic experience for her characters and for her audience. In Topdog/Underdog, Booth and Lincoln fight to exert control over the other. Each maintains that his masculine virility and intellectual prowess far surpasses that of the other. 
As the play progresses, we learn that Booth wants Lincoln to take up card hustling again. He points to Lincoln's lack of job security as the reason to take up a more lucrative career. For his part, Lincoln is wary of falling back on this unscrupulous way of earning a living. His previous partner, Lonny, had been shot during one of their card hustling ventures, and he remembers vowing that he would never go back to that life again after that tragedy. Booth continues to pressure Lincoln, maintaining that their combined card-hustling skills could be their golden ticket to life-long wealth. He uses sly threats and boastful proclamations about a $500 inheritance to goad Lincoln into action.
Booth reminds Lincoln that it has always been the two of them against the world. Even after their parents abandoned them, the two have always survived by looking out for each other and working together. Booth's demands take on new urgency after Lincoln loses his job. Additionally, he slyly hints that he and Grace will marry soon and that the newlyweds will need the apartment to themselves. Booth's incessant wheedling soon wears Lincoln down, and he decides to practice his game once more.
The play ends on a tragic note, with Booth reprising the role of his historical namesake. As John Wilkes Booth assassinated President Abraham Lincoln, Booth kills his own brother, Lincoln. Booth's last act makes him the top-dog by default, but he derives very little comfort from his Pyrrhic victory. The conflict is resolved but only on a superficial level. Booth will never work with his brother again.
 
https://www.nytimes.com/2001/07/22/theater/theater-this-time-the-shock-is-her-turn-toward-naturalism.html