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

TL;DR

This work advances the understanding of large induced substructures in planar graphs by linking the Albertson-Berman induced-forest conjecture with bounded-treewidth and outerplanarity concepts. It proves a strengthened partition theorem for $k$-trees, yielding large induced subgraphs of bounded treewidth and translating to concrete bounds for planar graphs; it also analyzes outerplanarity invariants $s_o$, $s_{o'}$, and $s_{K_4}$, showing that their gaps can be arbitrarily large while providing sharp linear upper bounds. The authors construct explicit triangulations to demonstrate unbounded differences between these invariants and establish tight bounds in the $K_4$-minor free setting, including a key lemma guaranteeing large induced outerplanar subgraphs. A central contribution is a counterexample family of plane Eulerian triangulations that disproves a broad connected-domination implication, clarifying the limits of that approach and motivating open questions about optimal constants in related conjectures and inequalities.

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.

Figures (15)

• Figure 1: Relations among the conjectures.
• Figure 2: The graph $R'$.
• Figure 3: The graph $D_k$, where a copy of $R'$ is embedded into each shaded triangle.
• Figure 4: The graph $I$, where the four vertices enclosed in circles form the set $S$.
• Figure 5: The graph $G_k$, where a copy of $I$ is embedded into each shaded triangle.

Theorems & Definitions (36)

• Conjecture 1: Albertson and Berman AB79
• Theorem 2: Hosono H90
• Theorem 3: Le Le18
• Theorem 4: Borodin B79
• Theorem 5: Dross et al. DMP19
• Theorem 6: Chappell and Pelsmajer CP13
• Theorem 1
• Conjecture 7: Borradaile et al. BLS17
• Conjecture 3
• Theorem 8: Angelini et al. AEFG16