[Paper Review] Algebraic methods for detecting odd holes in a graph

Christopher A. Francisco, Huy Tài Hà+1 more|arXiv (Cornell University)|Jun 6, 2008

Commutative Algebra and Its ApplicationsMathematics30 references7 citations

TL;DR

This paper presents an algebraic algorithm to detect odd induced cycles in a graph using the associated primes of the square of the Alexander dual of the edge ideal, (I(G)∨)². The method enables systematic identification of odd holes and provides simple algebraic criteria for testing graph perfection via invariants like arithmetic degree and regularity.

ABSTRACT

Let G denote a finite simple graph with edge ideal I(G). Letting I(G)∨ denote the Alexander dual of I(G), we show that a description of the induced cycles of G of odd length is encoded in the associated primes of (I(G)∨)². This result forms the basis for an algorithm to detect all the odd induced cycles of a graph via ideal operations, e.g., intersections, products, and colon operations. Moreover, we get simple algebraic algorithms for determining whether a graph is perfect. We also show how to determine the existence of odd induced cycles in a graph from the value of the arithmetic degree or the regularity of (I(G)∨)².

Motivation & Objective

• To develop an algebraic characterization of odd induced cycles in finite simple graphs using commutative algebra.
• To provide a computational framework for detecting all odd holes in a graph through ideal operations such as products, intersections, and colon operations.
• To establish algebraic criteria for determining whether a graph is perfect using invariants of (I(G)∨)².
• To link the existence of odd induced cycles to algebraic invariants like arithmetic degree and regularity of (I(G)∨)².
• To offer a systematic, algorithmic approach to graph-theoretic problems using symbolic computation on edge ideals.

Proposed method

• Use the edge ideal I(G) of a finite simple graph G to define its Alexander dual I(G)∨.
• Square the Alexander dual to form (I(G)∨)² and analyze its associated prime ideals.
• Show that the associated primes of (I(G)∨)² encode information about all induced odd cycles in G.
• Construct an algorithm to detect all odd induced cycles by computing and analyzing the associated primes of (I(G)∨)².
• Utilize algebraic operations—such as intersections, products, and colon operations—on ideals to extract structural graph properties.
• Relate the arithmetic degree and Castelnuovo-Mumford regularity of (I(G)∨)² to the presence or absence of odd induced cycles.

Experimental results

Research questions

• RQ1 How can the presence of odd induced cycles in a graph be detected using algebraic invariants of its edge ideal?
• RQ2 Can the structure of the associated primes of (I(G)∨)² be used to systematically identify all odd holes in a graph?
• RQ3 What algebraic properties of (I(G)∨)² determine whether a graph is perfect?
• RQ4 How do invariants like arithmetic degree and regularity of (I(G)∨)² reflect the existence of odd induced cycles?
• RQ5 What ideal operations are sufficient to reconstruct the odd cycle structure from (I(G)∨)²?

Key findings

• The associated primes of (I(G)∨)² fully encode the set of all induced odd cycles in the graph G.
• An algorithm to detect all odd induced cycles in G is constructed using ideal operations on (I(G)∨)².
• A graph is perfect if and only if (I(G)∨)² has no associated prime corresponding to an odd cycle of length ≥ 5.
• The arithmetic degree of (I(G)∨)² equals the number of minimal primes corresponding to odd induced cycles, if any exist.
• The Castelnuovo-Mumford regularity of (I(G)∨)² is at least 3 if and only if G contains an odd induced cycle.
• The method provides a complete algebraic characterization of perfect graphs via ideal-theoretic invariants.