Isolated toughness for path factors in networks
Sufang Wang, Wei Zhang · RAIRO - Operations Research · 2022
Let ℋ be a set of connected graphs. Then an ℋ-factor is a spanning subgraph ofG, whose every connected component is isomorphic to a member of the set ℋ. An ℋ-factor is called a path factor if every member of the set ℋ is a path. Letk ≥ 2 be an integer. By aP≥k-factor we mean a path factor in which each component path admits at leastkvertices. A graphGis called a (P≥k, n)-factor-critical covered graph if for anyW ⊆ V(G) with |W| = nand anye ∈ E(G − W),G− Whas aP≥k-factor coveringe. In this article, we verify that (1) an (n + λ + 2)-connected graphGis a (P≥2, n)-factor-critical covered graph if its isolated toughnessI(G) >n+λ+2/2λ+3, wherenandλare two nonnegative integers; (2) an (n+ λ + 2)-connected graphGis a (P≥3, n)-factor-critical covered graph if its isolated toughnessI(G) >n+3λ+5/2λ+3, wherenandλbe two nonnegative integers.