Feedback Vertex Set and Even Cycle Transversal for $H$-Free Graphs: Finding Large Block Graphs

Abstract

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph.In particular, we prove that both problems are polynomial-time solvable for sP3-free graphs for every integer s ≥ 1; here, the graph sP3 denotes the disjoint union of s paths on three vertices. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs. ACM Subject Classification Mathematics of computing → Graph algorithmsKeywords and phrases Feedback vertex set, even cycle transversal, odd cactus, forest, block

Computing Weighted Subset Transversals in H-Free Graphs

“…Both Subset Feedback Vertex Set and Subset Odd Cycle Transversal are polynomial-time solvable on H-free graphs if H = P 4 or H = sP 1 + P 3 [3,12]. Additionally, Feedback Vertex Set is polynomial-time solvable on P 5 -free graphs [1] and sP 3 -free graphs for every integer s ≥ 1 [11], and both Feedback Vertex Set and Odd Cycle Transversal are polynomial-time solvable on sP 2 -free graphs for every s ≥ 1 [4].…”

Computing Weighted Subset Odd Cycle Transversals in H-free graphs

“…Abrishami et al [1] proved that Weighted Feedback Vertex Set is polynomial-time solvable on P 5free graphs. Paesani et al [25] recently extended this result to (sP 1 + P 5 )-free graphs for every s ≥ 1 [24] and also proved that Weighted Feedback Vertex Set is polynomialtime solvable on sP 3 -free graphs for every s ≥ 1 [24]. In another recent paper [15], Gartland et al proved that Weighted Feedback Vertex Set is quasipolynomial-time solvable for P t -free graphs for every t ≥ 1.…”

“…We note finally that that there are other similar transversal problems that have been studied, but their complexity classifications on H-free graphs have not been settled: (Subset) Even Cycle Transversal [18,21,24], for example. Versions of the transversal problems that we have considered that have the additional constraint that the transversal must induce either a connected graph or an independent set have also been studied for H-free graphs [4,8,13,17].…”

MIP formulations for induced graph optimization problems: a tutorial

“…These include integer programming (Brunetta et al., 2000; Melo and Ribeiro, 2022), metaheuristics (Carrabs et al., 2011; Melo et al., 2021a), parameterized complexity (Burrage et al., 2006; Dehne et al., 2007; Kociumaka and Pilipczuk, 2014; Cao et al., 2015; Bergougnoux and Kanté, 2019; Jaffke et al., 2020b; Bergougnoux and Kanté, 2021), approximation algorithms (Chudak et al., 1998; Bafna et al., 1999), and exact exponential combinatorial algorithms (Fomin et al., 2006; Razgon, 2006; Fomin et al., 2008). However, polynomial‐time algorithms are known for obtaining induced forests in graphs with certain structures (Yannakakis and Gavril, 1987; Gavril, 2008; Abrishami et al., 2021; Paesani et al., 2022). In addition, graph theoretical properties regarding feedback vertex sets and induced forests were considered by several authors (Hansen et al., 2009; Dross et al., 2016; Kelly and Liu, 2017; Shi and Xu, 2017; Dross et al., 2019a, 2019b; Krivoshapko and Zhukovskii, 2021).…”