Backward Arcs in Hamilton Oriented Cycles and Paths in Directed Graphs with Independence Number Two

2026-03-24

AI Summary

This study addresses the problem of minimizing the number of backward arcs in Hamiltonian directed cycles and paths in digraphs with independence number at most two. Through structural analysis, connectivity theory, and constructive methods for directed paths, the authors establish—without relying on minimum degree conditions—that every 2-connected such digraph contains a Hamiltonian directed cycle with at most five backward arcs, while every 1-connected instance admits a Hamiltonian directed path with at most two backward arcs. This work provides tight upper bounds on the number of backward arcs in Hamiltonian structures for digraphs of independence number two, thereby advancing beyond traditional approaches that depend on degree-based assumptions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

In a digraph $D=(V,A)$, an oriented path is a sequence $P=x_1x_2\dots x_p$ of distinct vertices such that either $x_ix_{i+1}\in A$ or $x_{i+1}x_{i}\in A$ or both for every $i\in [p-1]$. If $x_ix_{i+1}\in A$ in $P$, then $x_ix_{i+1}$ is a forward arc of $P$; otherwise, $x_{i+1}x_{i}$ is a backward arc. The independence number $α(D)$ is the maximum integer $p$ such that $D$ has a set of $p$ vertices where there is no arc between any pair of vertices. A digraph is $k$-connected if its underlying undirected graph is $k$-connected. Freschi and Lo (JCT-B 2024) proved that every $n$-vertex oriented graph with minimum degree $δ\ge n/2$ has a Hamilton oriented cycle with at most $n-δ$ backward arcs. We prove that every 2-connected digraph $D$ with $α(D)\le 2$ has a Hamilton oriented cycle with at most five backward arcs, and every 1-connected digraph $D$ with $α(D)\le 2$ has a Hamilton oriented path with at most two backward arcs.

Problem

• Hamilton oriented cycle
• backward arcs
• independence number
• directed graphs
• Hamilton oriented path

Innovation

• Hamilton oriented cycle
• backward arc
• independence number
• directed graph
• connectivity