Hedge Connectivity without Hedge Overlaps

TL;DR

This paper investigates hedge connectivity in hedge graphs without overlaps, exploring its properties and computational challenges, and provides new insights into its algorithmic solutions.

Contribution

It introduces a detailed study of hedge graph connectivity without overlaps, addressing a long-standing open problem and proposing new combinatorial and algorithmic approaches.

Findings

Analyzed extremal conditions of hedge connectivity.

Developed hedge contraction operations for algorithmic progress.

Provided insights into polynomial-time solvability.

Abstract

Connectivity is a central notion of graph theory and plays an important role in graph algorithm design and applications. With emerging new applications in networks, a new type of graph connectivity problem has been getting more attention--hedge connectivity. In this paper, we consider the model of hedge graphs without hedge overlaps, where edges are partitioned into subsets called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. This model is more general than the hypergraph, which brings new computational challenges. It has been a long open problem whether this problem is solvable in polynomial time. In this paper, we study the combinatorial properties of hedge graph connectivity without hedge overlaps, based on its extremal conditions as well as hedge contraction operations, which provide new insights into its algorithmic progress.