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Survey of Results on the ModPath and ModCycle Problems

Antoine Amarilli

TL;DR

This survey compiles known complexity results for $\mathsf{ModPath}_{p,q}$ and $\mathsf{ModCycle}_{p,q}$, which ask for simple paths or cycles with lengths congruent to $p$ modulo $q$ in graphs. It presents a network of reductions between path and cycle variants, analyzes behavior across graph classes (directed, bounded-treewidth, high connectivity), and summarizes both exact and parity-based results, special-case open questions, and extensions to multiple disjoint structures. The key contributions include a taxonomy of tractability for fixed $q$, the role of graph structure (treewidth, degree, and connectivity) in enabling PTIME algorithms, and connections to parity, Erdős–Pósa properties, and graph decompositions. These findings guide algorithm design for constrained-length subgraph problems and illuminate how structural graph properties influence the feasibility of modular-length constraints.

Abstract

This note summarizes the state of what is known about the tractability of the problem ModPath, which asks if an input undirected graph contains a simple st-path whose length satisfies modulo constraints. We also consider the problem ModCycle, which asks for the existence of a simple cycle subject to such constraints. We also discuss the status of these problems on directed graphs, and on restricted classes of graphs. We explain connections to the problem variant asking for a constant vertex-disjoint number of such paths or cycles, and discuss links to other related work.

Survey of Results on the ModPath and ModCycle Problems

TL;DR

This survey compiles known complexity results for and , which ask for simple paths or cycles with lengths congruent to modulo in graphs. It presents a network of reductions between path and cycle variants, analyzes behavior across graph classes (directed, bounded-treewidth, high connectivity), and summarizes both exact and parity-based results, special-case open questions, and extensions to multiple disjoint structures. The key contributions include a taxonomy of tractability for fixed , the role of graph structure (treewidth, degree, and connectivity) in enabling PTIME algorithms, and connections to parity, Erdős–Pósa properties, and graph decompositions. These findings guide algorithm design for constrained-length subgraph problems and illuminate how structural graph properties influence the feasibility of modular-length constraints.

Abstract

This note summarizes the state of what is known about the tractability of the problem ModPath, which asks if an input undirected graph contains a simple st-path whose length satisfies modulo constraints. We also consider the problem ModCycle, which asks for the existence of a simple cycle subject to such constraints. We also discuss the status of these problems on directed graphs, and on restricted classes of graphs. We explain connections to the problem variant asking for a constant vertex-disjoint number of such paths or cycles, and discuss links to other related work.
Paper Structure (9 sections, 12 theorems)

This paper contains 9 sections, 12 theorems.

Table of Contents

  1. Definition
  2. Easy reductions
  3. On directed graphs
  4. Testing if all moduli are the same, and the case of modulo 2
  5. Cycles when the remainder is zero
  6. Bounded-treewidth graphs
  7. Constraints on degree and connectivity
  8. Multiple paths or multiple cycles
  9. Other related work

Key Result

Proposition 2.1

For any integers $p$ and $q>0$, the problem $\mathsf{ModCycle}_{p,q}$ reduces in PTIME (with a Turing reduction) to the problem $\mathsf{ModPath}_{p-1,q}$.

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 12 more