Rooted Cycle Bases

TL;DR

This paper introduces the concept of rooted cycle bases in graphs, characterizes their existence, and provides efficient algorithms for constructing and optimizing them, including solutions for minimum weight and fixed-parameter tractable cases.

Contribution

It defines rooted cycle bases, characterizes their existence conditions, and develops algorithms for their construction and optimization, including polynomial-time solutions and fixed-parameter algorithms.

Findings

Rooted cycle bases exist iff the root edge is in the 2-core and the 2-core is 2-vertex-connected.

Constructing a rooted cycle basis can be done efficiently.

Finding a minimum weight rooted cycle basis is polynomial-time solvable.

Abstract

A cycle basis in an undirected graph is a minimal set of simple cycles whose symmetric differences include all Eulerian subgraphs of the given graph. We define a rooted cycle basis to be a cycle basis in which all cycles contain a specified root edge, and we investigate the algorithmic problem of constructing rooted cycle bases. We show that a given graph has a rooted cycle basis if and only if the root edge belongs to its 2-core and the 2-core is 2-vertex-connected, and that constructing such a basis can be performed efficiently. We show that in an unweighted or positively weighted graph, it is possible to find the minimum weight rooted cycle basis in polynomial time. Additionally, we show that it is NP-complete to find a fundamental rooted cycle basis (a rooted cycle basis in which each cycle is formed by combining paths in a fixed spanning tree with a single additional edge) but that the problem can be solved by a fixed-parameter-tractable algorithm when parameterized by clique-width.