[Paper Review] Cohen-Macaulay graphs arising from digraphs

Giuseppa Carrà Ferro, Daniela Ferrarello|ArXiv.org|Mar 14, 2007

Commutative Algebra and Its Applications19 references8 citations

TL;DR

This paper establishes a one-to-one correspondence between transitive directed graphs (digraphs) and Cohen-Macaulay (CM) bipartite undirected graphs via a construction that maps digraphs to undirected graphs with perfect matchings. The key result is that a digraph is transitive if and only if its associated undirected graph is Cohen-Macaulay, linking graph transitivity to algebraic properties of toric ideals and providing a bridge between directed graph theory and commutative algebra.

ABSTRACT

In this paper we show a correspondence between directed graphs and bipartite undirected graphs with a perfect matching, that allows to study properties of directed graphs through the properties of the corresponding undirected graphs. In particular it is shown that a directed graph is transitive iff a corresponding undirected graph is Cohen-Macaulay.

Motivation & Objective

• To establish a correspondence between directed graphs and bipartite undirected graphs with perfect matchings.
• To characterize transitive digraphs through the Cohen-Macaulay property of their associated undirected graphs.
• To provide a bridge between graph-theoretic properties of digraphs (e.g., transitivity, acyclicity) and algebraic properties (e.g., Cohen-Macaulayness, toric ideals).
• To enable the use of both graph algorithms and computational algebra tools for verifying properties of digraphs via their associated graphs.

Proposed method

• Constructing an undirected bipartite graph $H_G$ from a digraph $G$ using a vertex set extension with source and sink vertices.
• Defining a sink-source graph $K_G$ from $G$'s sources and sinks, which forms a subgraph of $H_G$.
• Using toric ideals and Gröbner basis theory to analyze the algebraic structure of $H_G$, particularly focusing on binomial ideals and initial ideals.
• Applying Hall’s theorem and matching theory to prove the existence and uniqueness of perfect matchings in $H_G$ when $G$ is transitive.
• Verifying that the number of edges in $H_G$ matches the maximal possible for a Cohen-Macaulay bipartite graph.
• Using topological sorting to order vertices in $G$ such that edges in the perfect matching of $H_G$ correspond to left-to-right directed paths in $G$.

Experimental results

Research questions

• RQ1 Under what conditions is the undirected graph $H_G$ associated with a digraph $G$ Cohen-Macaulay?
• RQ2 What is the precise correspondence between transitive digraphs and Cohen-Macaulay bipartite graphs?
• RQ3 How do the structural properties of $G$ (e.g., sources, sinks, acyclicity) relate to the combinatorial and algebraic properties of $H_G$?
• RQ4 Can the existence of a unique perfect matching in $H_G$ be used to characterize transitivity in $G$?
• RQ5 What is the role of the sink-source graph $K_G$ in determining the Cohen-Macaulay property of $H_G$?

Key findings

• A digraph $G$ is transitive if and only if its associated undirected graph $H_G$ is Cohen-Macaulay.
• The associated graph $H_G$ is a bipartite graph with a unique perfect matching if and only if $G$ is transitive.
• The number of edges in $H_G$ is bounded by $\frac{(n-s+1)(n-s)}{2}$, where $s$ is the number of sources and sinks in $G$, and this bound is achieved precisely when $H_G$ is Cohen-Macaulay.
• The unique perfect matching in $H_G$ corresponds to a set of $s$ vertex-disjoint directed paths in $G$, covering all vertices and connecting sources to sinks.
• The construction ensures that $H_G$ is unmixed and has a unique minimal vertex cover, which is a necessary condition for Cohen-Macaulayness.
• The topological ordering of $G$'s vertices can be chosen such that edges in the perfect matching of $H_G$ correspond to consecutive vertices in the ordering, ensuring all edges in $G$ are directed left to right.