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Direct Product Primality Testing of Graphs is GI-hard

Luca Calderoni, Luciano Margara, Moreno Marzolla

TL;DR

This work analyzes graph primality testing under the direct (Kronecker) product and shows that, for possibly unconnected, nonbipartite graphs, primality and factorization are GI-hard. By constructing a polynomial-time, isomorphism-preserving reduction from graph isomorphism to compositeness on a carefully defined class $\Theta$, and then extending to general connected graphs via a transformation $f$, the authors establish that a polynomial-time primality test would yield a polynomial-time GI algorithm. The results highlight the critical impact of connectedness on the complexity of direct-product graph factorization and provide a concrete bridge between GI and primality problems. Open questions remain about primality for bipartite-connected graphs and about practical heuristics for large, possibly disconnected graphs.

Abstract

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem.

Direct Product Primality Testing of Graphs is GI-hard

TL;DR

This work analyzes graph primality testing under the direct (Kronecker) product and shows that, for possibly unconnected, nonbipartite graphs, primality and factorization are GI-hard. By constructing a polynomial-time, isomorphism-preserving reduction from graph isomorphism to compositeness on a carefully defined class , and then extending to general connected graphs via a transformation , the authors establish that a polynomial-time primality test would yield a polynomial-time GI algorithm. The results highlight the critical impact of connectedness on the complexity of direct-product graph factorization and provide a concrete bridge between GI and primality problems. Open questions remain about primality for bipartite-connected graphs and about practical heuristics for large, possibly disconnected graphs.

Abstract

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem.

Paper Structure

This paper contains 9 sections, 6 theorems, 36 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and Basic Definitions
  3. Related works
  4. Direct product
  5. Cartesian product
  6. Strong product
  7. Lexicographic product
  8. Main Results
  9. Conclusions

Key Result

Lemma 4.6

Let $G_1, G_2$ be undirected, connected graphs with $n$ nodes. Let $\mathbf{M}_1=\mathbf{Adj(\mathnormal{G_1})}$ and $\mathbf{M}_2=\mathbf{Adj(\mathnormal{G_2})}$. Let $\mathbf{M}$ denote the adjacency matrix of the disjoint union $G = G_1 \cup G_2$. Without loss of generality, we may write $\mathbf Then, $G_1$ and $G_2$ are isomorphic ($G_1 \mathop{\mathrm{\simeq}}\nolimits G_2$) if and only if t

Figures (3)

  • Figure 1: Example of the different types of graph products.
  • Figure 2: $G_2$ is a connected graph with a prime number of nodes. $G_3$ is the disjoint union of two copies of $G_2$. $G_3$ admits two different factorizations, namely, $G_1 \mathop{\mathrm{\times}}\nolimits G_2$ and $D_2 \mathop{\mathrm{\times}}\nolimits G_2$.
  • Figure 3: $G_3$ is the disjoint union of two connected graphs with the same number of nodes and edges. $G_3$ is the is the direct product of $G_1$ and $G_2$. $G_3$ admits as a factor $G_1$ which is a graph with two nodes different from $D_2$. $G_3$ does not admit $D_2$ as a factor.

Theorems & Definitions (21)

  • Definition 2.1: Cartesian product
  • Definition 2.2: Direct product
  • Definition 2.3: Strong product
  • Definition 2.4: Lexicographic product
  • Definition 4.1: GI[$S\space$]
  • Definition 4.2: Primality[$S\space$]
  • Definition 4.3: Compositeness[$S\space$]
  • Definition 4.4: $GI$-hard problem
  • Lemma 4.6
  • proof
  • ...and 11 more