Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n × n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 n 6 and n − 3 for n 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n 5, the domination number is at most n − 1 for n 4, and the independent domination number is at most n for n 5.

The classic n-queens problem asks for placements of just n mutually non-attacking queens on an n × n board. By adding enough pawns, we can arrange to fill roughly one-quarter of the board with mutually non-attacking queens. How many pawns do we need? We discuss that question for square boards as well as rectangular m × n boards.

A dragon king is a shogi piece that moves any number of squares vertically or horizontally or one square diagonally but does not move through or jump over other pieces. We construct infinite families of solutions to the n + k dragon kings problem of placing k pawns and n + k mutually nonattacking dragon kings on an n×n board, including solutions symmetric with respect to quarter-turn or half-turn rotations, solutions symmetric with respect to one or two diagonal reections, and solutions not symmetric with respect to any nontrivial rotation or reection. We show that an n + k dragon kings solution exists whenever n > k + 5 and that, given some extra conditions, symmetric solutions exist for n > 2k + 5.

We define the queens (resp., rooks) diameter-separation number to be the minimum number of pawns for which some placement of those pawns on an n × n board produces a board with a queens graph (resp., rooks graph) with a desired diameter d. We determine these numbers for some small values of d.