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