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

Abstract

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

“…We are now ready to prove Theorem 2, which we restate below. The proof mimics the proof of [24] and as mentioned at the start of this section, we include it only for reviewing purposes.…”

“…We compute a minimum vertex cover S ′ of G − (D ∪ W ) in polynomial time by Theorem 1. To be more precise, this takes O(n s+14 ) time by using the same arguments as in the proof of Lemma 8 (see [24]). Clearly S ′ ∪ D ∪ W is a vertex cover of G. As D is a connected dominating set, S ′ ∪ D ∪ W is even a connected vertex cover of G. Let S ∅ = S ′ ∪ D ∪ W .…”

“…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]. Their complexity on P r -free graphs is unknown for r ≥ 7 and r ≥ 6, respectively.…”

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

“…Recall that Grzesik et al [17] proved that Vertex Cover is polynomial-time solvable for P 6 -free graphs. Using the folklore trick mentioned in Remark 2 (see also, for example, [20,24]) their result can be formulated as follows.…”

“…For each s ≥ 1, Vertex Cover (by combining the results of [1,31]) and Connected Vertex Cover [8] are polynomial-time solvable for sP 2 -free graphs. 3 Moreover, Vertex Cover is also polynomial-time solvable for (sP 1 + P 6 )-free graphs, for every s ≥ 0 [17], whereas Connected Vertex Cover is so for (sP 1 + P 5 )-free graphs [20]. Their complexity for P r -free graphs is unknown for r ≥ 7 and r ≥ 6, respectively.…”