insight - Graph Theory - # Computational Complexity of Well-Covered and Generalized Well-Covered Graphs

Core Concepts

The core message of this article is to investigate the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs. The authors establish several complexity results, including showing that recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, determining the precise complexity of recognizing 1-extendable (Es) graphs as Θp2-complete, and providing a linear-time algorithm to decide if a chordal graph is 1-extendable.

Abstract

The article focuses on the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs.
Key highlights:
Recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, resolving open problems.
Recognizing 1-extendable (Es) graphs is Θp2-complete, closing the complexity gap.
A linear-time algorithm is provided to decide if a chordal graph is 1-extendable, addressing an open question.
The complexity of recognizing well-covered triangle-free graphs and co-well-covered graphs remains open.
The authors first introduce the Wk hierarchy, where a graph G is Wk if for any k pairwise disjoint independent sets, there exist k pairwise disjoint maximum independent sets containing them. They show that recognizing Wk+1 graphs is coNP-complete, even when the input graph is Wk or Es.
Next, the authors investigate the complexity of recognizing Es graphs, where a graph is Es if every independent set of size at most s is contained in a maximum independent set. They prove that recognizing Es graphs is Θp2-complete, even when the input graph is Es-1.
For chordal graphs, the authors provide a linear-time algorithm to decide if a chordal graph is 1-extendable, by characterizing 1-extendable chordal graphs as those that can be partitioned into maximal cliques. They also show that recognizing Es chordal graphs is coW[2]-hard when parameterized by s.
The article concludes by highlighting two open problems: the complexity of recognizing well-covered triangle-free graphs and co-well-covered graphs.

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Deeper Inquiries

The implications of the Θp2-completeness of recognizing 1-extendable graphs are significant in the design of efficient algorithms for related problems in graph theory and optimization. This complexity result indicates that the problem of recognizing 1-extendable graphs is among the hardest problems in the polynomial hierarchy with access to a SAT oracle. As a result, it highlights the inherent computational complexity and challenges associated with this specific graph class.
In practical terms, the Θp2-completeness implies that recognizing 1-extendable graphs cannot be efficiently solved using standard polynomial-time algorithms without leveraging the power of a SAT oracle. This complexity result underscores the need for advanced algorithmic techniques, possibly involving SAT solvers or other sophisticated methods, to tackle problems related to 1-extendable graphs effectively.
Furthermore, the insights gained from understanding the computational complexity of recognizing 1-extendable graphs can guide researchers and algorithm designers in developing more efficient algorithms for optimization problems that involve similar graph structures. By leveraging the knowledge of the complexity landscape, algorithm designers can tailor their approaches to handle the intricacies of 1-extendable graphs and potentially extend these techniques to address other challenging graph optimization problems.

The techniques used to establish the coNP-hardness of recognizing Wk graphs can potentially be extended to other generalized well-covered graph classes or related problems in graph theory. The approach of reduction from well-known NP-hard problems, such as Exact MIS or Minimum Dominating Set, can serve as a foundational strategy for proving the computational complexity of recognizing various graph classes.
By adapting the reduction techniques and problem formulations to suit the specific characteristics of different graph classes, researchers can explore the coNP-completeness or hardness of recognition problems for a wide range of graph structures beyond Wk graphs. This extension of techniques can provide valuable insights into the complexity landscape of generalized well-covered graph classes and related optimization problems.
Moreover, the foundational principles and methodologies employed in proving the coNP-hardness of recognizing Wk graphs can serve as a blueprint for investigating the computational complexity of other graph recognition problems. By building upon these established techniques and tailoring them to suit the unique properties of different graph classes, researchers can uncover new insights into the hardness of recognition tasks in graph theory.

The characterization of 1-extendable chordal graphs offers valuable insights into the structural properties of these graphs and provides a systematic way to understand their unique characteristics. By identifying a partition of the vertex set into maximal cliques as a defining feature of 1-extendable chordal graphs, researchers gain a deeper understanding of how these graphs are organized and interconnected.
These insights into the structure of 1-extendable chordal graphs can be leveraged to develop more efficient algorithms for solving related problems on chordal graphs. The characterization provides a clear framework for identifying key structural elements within 1-extendable chordal graphs, which can be utilized to optimize algorithm design and streamline computational processes.
Furthermore, the characterization of 1-extendable chordal graphs opens up avenues for exploring connections between different graph classes and understanding the relationships between various graph properties. By leveraging the insights gained from this characterization, researchers can potentially uncover new algorithmic approaches, problem-solving strategies, and optimization techniques for chordal graphs and related graph classes.

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