Graph-based network parameters and the existence of ℋ-factors in graphs

For a family of connected graphs [Formula: see text], an [Formula: see text]-factor is a spanning subgraph of a graph, whose connected components are isomorphic to graphs from the set [Formula: see text]. A graph [Formula: see text] is called an [Formula: see text]-factor avoidable graph if for any [Formula: see text], [Formula: see text] admits an [Formula: see text]-factor excluding [Formula: see text]. Furthermore, a graph [Formula: see text] is called an [Formula: see text]-factor critical avoidable graph if for every [Formula: see text] with [Formula: see text] is an [Formula: see text]-factor avoidable graph. Let [Formula: see text] be a positive integer with [Formula: see text], and write [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are two special families of trees. Some graph-based network parameters such as isolated toughness, binding number and degree sum, can directly measure the vulnerability of the network. In this paper, we use the these network parameters to study [Formula: see text]-factor critical avoidable graphs and [Formula: see text]-factor critical avoidable graphs, respectively.