On graphic parameters and component factor critical avoidable graphs

Source abstract

Graph theory, a pivotal subfield within modern mathematics, has experienced remarkable expansion. This growth can be attributed to its critical function in offering structural frameworks and essential instruments for computer science, communication network analysis, and combinatorial optimization challenges. By integrating methodologies from diverse mathematical disciplines, such as the probabilistic approach, linear algebra, group theory, and topology, graph theory has continuously enhanced its theoretical depth and practical applicability. Among the core research areas in graph theory, factor theory stands out as one of the earliest and most fundamental topics of investigation. Given a collection of connected graphs denoted as $\mathscr{H}$, an $\mathscr{H}$-factor of a graph is defined as a spanning subgraph where each connected component is isomorphic to an element of the set $\mathscr{H}$. A graph $G$ is called an $\mathscr{H}$-factor avoidable graph if for any $e\in E(G)$, $G$ admits an $\mathscr{H}$-factor excluding $e$. Furthermore, a graph $G$ is called an $(\mathscr{H},l)$-factor critical avoidable graph if for every $V'\subseteq V(G)$ with $|V'|=l, G-V'$ is an $\mathscr{H}$-factor avoidable graph. In this paper, we use some graph parameters such as toughness, isolated toughness, binding number and degree sum, to study path-factor critical avoidable graphs and star-factor critical avoidable graphs, respectively.