Cyclewidth: an Analogy of Treewidth

In this work we study cyclewidth, a generalization of treewidth and pathwidth to cycles. Although treewidth and pathwidth have been extensively studied, cyclewidth has only been mentioned in a single paper, to the best of the author’s knowledge. We prove that the decision problem of determining whether the cyclewidth of a graph is at most k is NP-complete. We also present an algorithm for finding the cyclewidth of cactus graphs and prove that the pathwidth of a given cactus graph is at most one greater than its cyclewidth. Additionally, we characterize edge-maximal graphs of pathwidth k and leave the problem of recognizing them unsolved.