On the matching arrangement of a graph, improper weight function problem and its application

TL;DR

This work investigates the matching arrangement $MA(G)$ of a graph and the complexity of the associated improper weight function problem. It applies the finite field method to compute the characteristic polynomials $ χ_{MA(G)}(x)$ for selected graph classes, notably deriving explicit formulas for even and odd cycles and a $K_3$-tail composition. It proves that the improper weight function problem is NP-complete via a polynomial reduction from 3-SAT and connects the count of proper weight functions to $ χ_{MA(G)}(q)$ under field extensions. Finally, it offers a cryptographic application—the Bolotnikov–Irmatov cryptosystem—where NP-completeness and an alternating weighted path variant underpin public-key security, with a linear operator disguising weights to mimic knapsack-based schemes. The results bridge hyperplane-arrangement theory with computational complexity and cryptography, providing both theoretical insights and a potential practical construction for secure communications.

Abstract

This article presents examples of an application of the finite field method for the computation of the characteristic polynomial of the matching arrangement of a graph. Weight functions on edges of a graph with weights from a finite field are divided into proper and improper functions in connection with proper colorings of vertices of the matching polytope of a graph. An improper weight function problem is introduced, a proof of its NP-completeness is presented, and a knapsack-like public key cryptosystem is constructed based on the improper weight function problem.

Figures (7)

• Figure 1: A cycle of length $n$
• Figure 2: Graph $G$
• Figure 3: Matrix of the operator $W$
• Figure 4: Subraph $Z_i$
• Figure 5: Subraph $Y_j$

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Proposition 3