An introduction to influence parameters of chessboard graphs
The influence of a vertex set S⊆V(G) is I(S)=∑v∈S(1+deg(v))=∑v∈S|N[v]|, which is the total amount of domination done by the vertices in S. Influence parameters include the efficient domination number F(G), the closed neighborhood order domination number W(G), the closed neighborhood order packing number P(g), and the redundance number R(G). For each chess piece the corresponding chessboard graphs have vertex sets that correspond to the squares of a board with two vertices adjacent if and only if the chess piece can go from one of the vertices' corresponding squares to the other in a single move.
This paper introduces the study of influence parameters for chessboard graphs.
Sinko A, Slater PJ. 2006. An introduction to influence parameters of chessboard graphs. Congressus Numerantium 172: 15-27.