Diameter Bounds for Friends-and-Strangers Graphs

2025-09-27

AI Summary

This work investigates the diameter of connected components in Friends-and-Strangers graphs. For the case where the underlying friendship graph is a star, we establish the first explicit upper bound of $O(n^4)$ on the diameter. We then generalize to arbitrary graphs: when both the friendship graph and the moving graph have sufficiently large minimum degree, we prove that the diameter is polynomially bounded. Furthermore, when both graphs are independent Erdős–Rényi random graphs with edge probability $p ge n^{-1/2+varepsilon}$, we show that the distance between any two configurations is polynomially bounded with high probability. Our approach integrates combinatorial graph theory, probabilistic analysis, and constructive path algorithms, yielding an efficient configuration transformation strategy. These results provide the first explicit polynomial diameter bounds for Friends-and-Strangers graphs and establish new theoretical foundations—and implementable algorithms—for polynomial-time solvability of reconfiguration problems such as sliding puzzles.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Security and Privacy: Large-scale security measurements

Abstract

Consider two $n$-vertex graphs $X$ and $Y$, where we interpret $X$ as a social network with edges representing friendships and $Y$ as a movement graph with edges representing adjacent positions. The friends-and-strangers graph $mathsf{FS}(X,Y)$ is a graph on the $n!$ permutations $V(X) o V(Y)$, where two configurations are adjacent if and only if one can be obtained from the other by swapping two friends located on adjacent positions. Friends-and-strangers graphs were first introduced by Defant and Kravitz, and generalize sliding puzzles as well as token swapping problems. Previous work has largely focused on their connectivity properties. In this paper, we study the diameter of the connected components of $mathsf{FS}(X, Y)$. Our main result shows that when the underlying friendship graph is a star with $n$ vertices, the friends-and-strangers graph has components of diameter $O(n^4)$. This implies, in particular, that sliding puzzles are always solvable in polynomially many moves. Our work also provides explicit efficient algorithms for finding these solutions. We then extend our results to general graphs in two ways. First, we show that the diameter is polynomially bounded when both the friendship and the movement graphs have large minimum degree. Second, when both the underlying graphs $X$ and $Y$ are Erdős-Rényi random graphs, we show that the distance between any pair of configurations is almost always polynomially bounded under certain conditions on the edge probabilities.

Problem

• Studying diameter bounds for friends-and-strangers graph components
• Analyzing polynomial diameter when friendship graph is a star
• Extending diameter bounds to general and random graph settings

Innovation

• Star graph analysis yields O(n^4) diameter bound
• Polynomial diameter proven for high-degree graphs
• Random graph configurations have polynomial distances