Contributions to conjectures in planar graphs: Induced Substructures, Treewidth, and Dominating Sets

Abstract: Two of the most prominent unresolved conjectures in graph theory, the Albertson-Berman conjecture and the Matheson-Tarjan conjecture, have been extensively studied by many researchers. (AB) Every planar graph of order $n$ has an induced forest of order at least $\frac{n}{2}$. (MT) Every plane triangulation of sufficiently large order $n$ has a dominating set of cardinality at most $\frac{n}{4}$. Although partial results and weaker bounds than those originally conjectured have been obtained, both problems remain open. To contribute to their resolution, various generalizations and variations of the original concepts have been investigated, such as total dominating set, induced linear forests, and others. In this paper, we clarify the relations among several notions related to these two major conjectures, such as connected domination and induced outerplanar subgraphs, etc., and survey the associated conjectures. We then provide counterexamples to some of these conjectures and establish the best bounds on the gap between the maximum orders of induced subgraphs under different structural conditions. In addition, we present a general upper bound on the order of induced subgraphs in terms of treewidth, a fundamental graph invariant.