Abstract
We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.
Highlights
Detecting the existence of cycles in graphs is a fundamental problem in graph theory
Using algebraic methods to study combinatorial structures and using combinatorial data to understand algebraic properties and invariants has evolved to be an active research topic in combinatorial commutative algebra in recent years. We continue this line of work and describe an algebraic algorithm to enumerate even circuits in an undirected graph; a circuit is a closed walk in which the edges are all distinct and a cycle is a closed walk in which the vertices are all distinct
We discuss an application of our work to the problem of finding directed cycles in a directed graph
Summary
Detecting the existence of cycles in graphs is a fundamental problem in graph theory (cf. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]). Graph theoretic algorithms exist to enumerate both odd and even cycles. In [16], the first author, together with Francisco and Van Tuyl, gave an algebraic algorithm to detect and exhibit all induced odd cycles in an undirected graph. We continue this line of work and describe an algebraic algorithm to enumerate even circuits in an undirected graph; a circuit is a closed walk in which the edges are all distinct and a cycle is a closed walk in which the vertices are all distinct. Our first theorem reads as follows, leaving unexplained terminology until
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