Minimum connected transversals in graphs: New hardness results and tractable cases using the price of connectivity

Abstract

Abstract. We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H. It is known that Connected Vertex Cover is NP-complete even for H-free graphs if H contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if H contains a cycle or claw. In the remaining case H is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for sP2-free graphs for every constant s ≥ 1. For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms.

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

“…For every s ≥ 1, Vertex Cover (by combining the results of [1,34]) and Connected Vertex Cover [10] are polynomial-time solvable on sP 2 -free graphs. 4 Moreover, Vertex Cover is also polynomial-time solvable on (sP 1 + P 6 )-free graphs, for every s ≥ 0 [20], as is the case for Connected Vertex Cover on (sP 1 + P 5 )-free graphs [24].…”

“…Grigoriev and Sitters [18] proved that Connected Feedback Vertex Set is NP-complete even on planar graphs with maximum degree 9. More recently, Chiarelli et al [10] proved that Connected Odd Cycle Transversal is NP-complete even on graphs of arbitrarily large girth and on line graphs.As all three decision problems and their connected variants are NP-complete, we can consider how to restrict the input to some special graph class in order to achieve tractability. Note that this approach is in line with the aforementioned results in the literature, where NP-completeness was proven on special graph classes.…”

“…Grigoriev and Sitters [18] proved that Connected Feedback Vertex Set is NP-complete even on planar graphs with maximum degree 9. More recently, Chiarelli et al [10] proved that Connected Odd Cycle Transversal is NP-complete even on graphs of arbitrarily large girth and on line graphs.…”

Connected Vertex Cover for $$(sP_1+P_5)$$-Free Graphs

“…This leads us to the research question of our paper: For which classes of graphs do the complexities of Vertex Cover and Connected Vertex Cover coincide? This question was addressed by Chiarelli et al [6] who considered classes of graphs characterized by a single forbidden induced subgraph H. Such graphs are called H-free. They observed that the results of Munaro [16] imply that Connected Vertex Cover is NP-complete for H-free graphs if H contains a cycle or a claw.…”

“…By using the concept of the price of connectivity [3,5,12], Chiarelli et al [6] proved that Connected Vertex Cover is polynomial-time solvable for sP 2free graphs for any integer s ≥ 1. For Vertex Cover this follows by combining two classical results [2,19] (as is well-known).…”