Question: Is it possible to discover a sequence of steps to checkmate using a king, bishop, and knight, while the enemy only has a king, on an 8x8 chessboard? Without loss of generality, assume that the king, bishop, and knight are white while the lone king is black.
I've tried to use graph theory to solve the problem, where I see all vertices that the white pieces fall on and which placements would allow the black king's graph to be a subset of the white pieces' graph.
I've also tried to create a bijection, where I let the possible moves of the pieces be functions while the graphs of the pieces are the ranges of the functions and the possible initial starting positions are the domains of the functions. While this seems to be a useful way to look at the problem, I'm not sure how to produce such functions. Perhaps looking at this in terms of functional equations could be helpful.
It would be preferable if the solution relied on mathematical concepts and proofs rather than just on logic or chess techniques since this problem is supposed to be solved using graph theory. However, any other methods apart from graph theory are also welcome!
You could create a state graph $G$ where each vertex $v$ represents a configuration of a black king, knight, bishop and a white king (by configuration I mean positions for each of them).
Then, add an edge $(u,v)$ between two vertices if it is possible to go from one cconfiguration to another in a single move.
A path from the initial configuration to any checkmate configuration is a sequence of steps to checkmate.
The graph is probably very very big though.