🤖 AI Summary
This paper addresses two longstanding open conjectures on planar graphs: the Albertson–Berman Conjecture (AB), asserting that every $n$-vertex planar graph contains an induced forest of order at least $n/2$, and the Matheson–Tarjan Conjecture (MT), stating that large $n$-vertex triangulations admit a dominating set of size at most $n/4$. We develop a unified analytical framework leveraging combinatorial graph theory, structural induction, treewidth parameterization, and planar embedding analysis. Our contributions include: (i) establishing novel logical connections between connected dominating sets, induced outerplanar subgraphs, and both conjectures; (ii) constructing critical counterexamples refuting several natural generalizations; (iii) determining optimal bounds on induced subgraph sizes under various structural constraints; (iv) proving that every graph $G$ admits an induced subgraph of order at most $mathrm{tw}(G)+1$, yielding a new upper-bound tool for planar graphs; and (v) deriving tight bounds for induced linear forests and outerplanar subgraphs—advancing the resolution of both conjectures substantially.
📝 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.