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

TL;DR

The paper develops a general framework to count connected vertex subsets in grid-like graphs. It first provides a transfer-matrix formula for $N(K_m\times P_n)$, enabling exact enumeration via $N(K_m\times P_n)=\sum_{k=1}^n (n-k+1)\cdot N_{m,k}$ with $N_{m,k}=\mathbf{u}\cdot T^{k-1}\mathbf{1}$. It then introduces a multistep-recurrence-based lower bound $N'_{m,n}$ for the number of connected sets in $K_m\times P_n$ that also hit every column, and proves a sharp upper bound $N(P_m\times P_n)\le \sum_{k=1}^n (N_{m,k}-N'_{m,k})(n-k+1)$, with equality for $m=3,4$. The sections 4.1 and 4.2 supply explicit, recursive schemes to compute $N(P_3\times P_n)$ and $N(P_4\times P_n)$, yielding exact counts for these special grid widths and illustrating the method's potential for larger widths. Overall, the work advances a structured, recurrence-based path toward Vince's question on formulas for grid-graph connected-set counts and offers practical enumeration for particular cases.

Abstract

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 $n$.

Figures (6)

• Figure 1: Examples (a), (b), (c), (d) for Lemma \ref{['lem1']} (1), (2), (3) (i), (3) (ii), respectively, where $C$ is the set of solid vertices
• Figure 2: An example for the illustration of Lemma \ref{['lem1']} (3.3), where the graph on the left is an example for Lemma \ref{['lem1']} (3) (iii) with $C$ the set of solid vertices
• Figure 3: An example of the case that Lemma \ref{['lem1']} can not deal with, where $C$ is the set of solid vertices
• Figure 4: The graph $K_{3}\times P_{n}$
• Figure 5: An example

Theorems & Definitions (16)

• Theorem 1.2
• Corollary 2.1
• Corollary 2.2
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Lemma 4.1
• proof
• Lemma 4.2
• proof