[Paper Review] Shedding vertices of vertex decomposable graphs

Jonathan M. Baker, Kevin N. Vander Meulen+1 more|arXiv (Cornell University)|Jun 14, 2016

Commutative Algebra and Its Applications19 references3 citations

TL;DR

This paper investigates the relationship between shedding vertices and dominating sets in vertex decomposable well-covered graphs. It constructs three new infinite families of such graphs where the set of shedding vertices fails to be a dominating set, providing the first minimal counterexamples to a conjecture by Villarreal regarding Cohen-Macaulay graphs, and demonstrates that this property does not hold universally even for pure vertex decomposable graphs on 10 or fewer vertices.

ABSTRACT

We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen-Macaulay graphs.

Motivation & Objective

• To investigate whether the set of shedding vertices in pure vertex decomposable well-covered graphs always forms a dominating set.
• To challenge the prevailing intuition that shedding vertices dominate in known families of vertex decomposable graphs.
• To construct new infinite families of vertex decomposable well-covered graphs where shedding vertices do not form a dominating set.
• To provide minimal counterexamples to Villarreal's conjecture on Cohen-Macaulay graphs using these constructions.
• To establish computational evidence that for graphs with ≤10 vertices, Cohen-Macaulay and pure vertex decomposable graphs coincide, but the dominating set property for shedding vertices fails in some cases.

Proposed method

• The authors analyze known families of vertex decomposable well-covered graphs (e.g., bipartite, chordal, Cameron-Walker, clique-whiskered, very well-covered, and girth ≥5 graphs) to verify that shedding vertices form a dominating set.
• They introduce three new infinite families of vertex decomposable well-covered graphs where the shedding vertex set is not a dominating set, using graph-theoretic constructions based on specific vertex attachments and edge additions.
• They employ computational tools (Macaulay2, Nauty, EdgeIdeals, SimplicialDecomposability) to verify vertex decomposability, well-coveredness, and Cohen-Macaulay properties for graphs up to 10 vertices.
• They use a vertex-duplication technique to generate larger graphs with the same non-dominating shedding set property from a base counterexample.
• They perform a systematic computer search over all connected graphs with ≤10 vertices to identify minimal counterexamples to the dominating set conjecture.
• They prove that for graphs with ≤10 vertices, the classes of Cohen-Macaulay and pure vertex decomposable graphs are identical, using computational verification.

Experimental results

Research questions

• RQ1 Is the set of shedding vertices always a dominating set in pure vertex decomposable well-covered graphs?
• RQ2 Can new infinite families of vertex decomposable well-covered graphs be constructed where the shedding vertices do not form a dominating set?
• RQ3 What is the smallest counterexample to Villarreal’s conjecture on Cohen-Macaulay graphs, and does it arise from a non-dominating shedding set?
• RQ4 Do standard constructions of vertex decomposable graphs (e.g., clique-whiskering, Cameron-Walker) always preserve the dominating set property of shedding vertices?
• RQ5 Are there graphs with ≤10 vertices that are Cohen-Macaulay but not pure vertex decomposable?

Key findings

• The set of shedding vertices is a dominating set for all known families of pure vertex decomposable well-covered graphs, including bipartite, chordal, Cameron-Walker, and very well-covered graphs.
• Three new infinite families of pure vertex decomposable well-covered graphs were constructed where the shedding vertex set is not a dominating set, disproving a broad conjecture about this property.
• The graph F₂ on nine vertices and 13 edges is the minimal counterexample to Villarreal’s conjecture, with the smallest number of edges among all such counterexamples.
• Among all connected graphs with ≤10 vertices, the classes of Cohen-Macaulay and pure vertex decomposable graphs are identical, as confirmed by computational search.
• The minimal counterexample F₂ has a shedding set that is not dominating, and for every vertex not in the shedding set, the deletion of that vertex results in a graph that is not well-covered or not Cohen-Macaulay.
• The paper confirms that the property of shedding vertices forming a dominating set does not hold universally, even in the smallest non-trivial cases, and provides a definitive counterexample to a long-standing conjecture.