Matching preclusion for $n$-grid graphs

TL;DR

This paper investigates the matching preclusion properties of n-grid graphs, determining the minimum edge sets whose removal prevents perfect matchings, with results depending on the graph's order parity.

Contribution

It characterizes the matching preclusion number and optimal sets for n-grid graphs, extending understanding of their fault tolerance based on order parity.

Findings

Even order n-grid graphs have matching preclusion number n with trivial optimal sets.

Odd order n-grid graphs have matching preclusion number n+1 with characterized optimal sets.

Results provide insights into fault tolerance of grid-like network topologies.

Abstract

A matching preclusion set of a graph is an edge set whose deletion results in a graph without perfect matching or almost perfect matching. The Cartesian product of $n$ paths is called an $n$-grid graph. In this paper, we study the matching preclusion problems for $n$-grid graphs and obtain the following results. If an $n$-grid graph has an even order, then it has the matching preclusion number $n$, and every optimal matching preclusion set is trivial. If the $n$-grid graph has an odd order, then it has the matching preclusion number $n+1$, and all the optimal matching preclusion sets are characterized.