Rooted Cycle Bases

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. Submitted: February 2017 Accepted: May 2017 Final: May 2017 Published: June 2017 Article type: Regular paper Communicated by: D. Wagner The work of the first author was supported by the National Science Foundation under Grants CCF-1228639, CCF-1618301, and CCF-1616248 and by the Office of Naval Research under Grant No. N00014-08-1-1015. A preliminary version of this work was presented at the 14th Algorithms and Data Structures Symposium (WADS 2015) and appears in the proceedings of that symposium, Lecture Notes in Computer Science 9214, Springer, 2015, pp. 339–350. E-mail addresses: [email protected] (David Eppstein) [email protected] (J. Michael McCarthy) [email protected] (Brian E. Parrish) 664 Eppstein, McCarthy, and Parrish Rooted Cycle Bases