A note on the minimum rank of graphs with given dominating induced subgraph

An induced subgraph of a graph \(G\) is said to be dominating if every vertex of \(G\) is at distance at most one from this subgraph. We investigate pairs \((G, F)\) where \(F\) is a non-singular dominating induced subgraph of \(G,\) and the rank of \(G\) (that is, the rank of its adjacency matrix) attains the minimum, i.e., equals the number of vertices in \(F\). It turns out that the inverse of the adjacency matrix of a non-singular path, half graph, or even cycle is the adjacency matrix of a related signed graph; here, a half graph refers to a connected chain graph with exactly one vertex in each cell. We exploit this property to give a complete characterization of graphs \(G\) paired with any of these graphs in the role of \(F\). The bipartite case is singled out. It occurs that every nonsingular \(F\) is paired with an infinite family of graphs \(G\), and their number is comparatively large even if we exclude the existence of the so-called twin vertices. The latter empirical observation is demonstrated through some examples.