A sharp upper bound for the number of connected sets in any grid graph

A connected set in a graph is a subset of vertices whose induced subgraph is connected. Although counting the number of connected sets in a graph is generally a \#P-complete problem, it remains an active area of research. In 2020, Vince posed the problem of finding a formula for the number of connected sets in the $(n\times n)$-grid graph. In this paper, we establish a sharp upper bound for the number of connected sets in any grid graph by using multistep recurrence formulas, which further derives enumeration formulas for the numbers of connected sets in $(3\times n)$- and $(4\times n)$-grid graphs, thus solving a special case of the general problem posed by Vince. In the process, we also determine the number of connected sets of $K_{m}\times P_{n}$ by employing the transfer matrix method, where $K_{m}\times P_{n}$ is the Cartesian product of the complete graph of order $m$ and the path of order $n$.