Contributions to conjectures on planar graphs: Induced Subgraphs, Treewidth, and Dominating Sets

TL;DR

This paper explores longstanding conjectures in planar graph theory, clarifies relationships among related concepts, and constructs counterexamples, advancing understanding of induced subgraphs, treewidth, and dominating sets.

Contribution

It clarifies relations among various notions related to planar graph conjectures and constructs counterexamples, providing new insights into induced subgraphs and domination parameters.

Findings

Constructed an infinite family of plane triangulations with high connected domination number

Provided a negative answer to a question about maxleaf number in plane triangulations

Obtained new results on induced subgraphs with bounded treewidth and outerplanar subgraphs

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 progress and weaker bounds are known, both conjectures remain unsolved. To shed further light on them, researchers have explored a variety of related notions and generalizations. In this paper, we clarify relations among several of these notions, most notably connected domination and induced outerplanar subgraphs, and investigate the corresponding open problems. Furthermore, we construct an infinite family of plane triangulations of order $n$ whose connected domination number exceeds $n/3$. This construction gives a negative answer to a question of Bradshaw et al. [SIAM J. Discrete Math. 36 (2022) 1416-1435], who asked whether the maxleaf number of every plane triangulation of order $n$ is at least $2n/3$. We also obtain new results on induced subgraphs with bounded treewidth and induced outerplanar subgraphs.