Induced planar Turán numbers

Abstract: The planar Turá number of a graph $F$ is the maximum number of edges an $n$-vertex $F$-free planar graph can have. We study the case where $F$ is forbidden as an induced subgraph, thereby introducing the induced planar Turá numbers. We will determine a sharp upper bound when $F$ is $Θ_4$, a $4$-cycle with a diagonal edge, and obtain exact extremal values in case $F$ is a path $P_k$ on $k$ vertices, for $k=3,4$ and $5$.

• The paper introduces induced planar Turán numbers and establishes sharp bounds for various forbidden induced subgraphs using triangulation analysis.
• It derives exact and asymptotic results for cases like Θ4 and short paths (P3, P4, P5) through precise combinatorial decompositions.
• Novel inductive techniques and block decompositions uncover threshold phenomena and clarify extremal behavior in planar graphs.

Induced Planar Turán Numbers: New Results and Techniques

Introduction

This paper investigates induced planar Turán numbers, introducing and precisely analyzing this variant of the classical Turán-type extremal problem under the simultaneous constraints of planarity and induced subgraph avoidance. For a fixed graph $F$, the induced planar Turán number, denoted $_(n, F^)$, is the maximum possible number of edges in an $n$-vertex planar graph that does not have an induced subgraph isomorphic to $F$. The authors provide exact and asymptotic results for various forbidden induced subgraphs $F$, most notably for small paths and for $\Theta_4$ (the four-cycle with a chord), leveraging a structural analysis of planar triangulations and a fine-grained combinatorial decomposition.

Structural Results and Key Extremal Bounds

The paper begins by rigorously formalizing the induced planar Turán number,

$$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$

and explores immediate consequences for planar graphs where $F$ is a clique, cycle, or a star with at least three leaves. In these settings, specific triangulations achieve the trivial bound of $3n-6$ edges, and key sufficient conditions for this extremal value are established:

• If $F$ contains $_(n, F^)$0, an induced cycle $_(n, F^)$1 ($_(n, F^)$2), an induced star $_(n, F^)$3 ($_(n, F^)$4), or an induced $_(n, F^)$5, then $_(n, F^)$6 for all $_(n, F^)$7.

For more delicate induced subgraphs, like the "theta" graphs $_(n, F^)$8 and short paths $_(n, F^)$9, the extremal functions are notably non-trivial. The paper isolates the set of cases where maximal planar triangulations fail to avoid an induced copy, and alternative constructions or bounds are necessary.

The $n$0-Free Case

The strongest new bound is derived for induced planar $n$1-free graphs. The critical lemma is that in such graphs, every triangular block—decompositions akin to facial subgraphs—has at most four vertices and must be isomorphic to $n$2, $n$3, $n$4, or $n$5. This constraint enables the authors to leverage block-by-block counting arguments and Euler's formula to show:

$n$6

where the bound is attained for all $n$7 through explicit recursive constructions with all blocks realized as $n$8 triangulations. The analysis is sharp and highlights structural ampleness—maximality of edge sets—compatible with induced $n$9-freeness.

Induced Planar Turán Numbers for Short Paths

Precise values for $F$0 are obtained for $F$1:

• For induced $F$2-free planar graphs, the extremal configuration is realized by disjoint unions of $F$3's and possibly one smaller clique, yielding

$F$4

with uniqueness of the extremal graph up to isomorphism.

• For $F$5, the threshold for achieving the maximal planar bound $F$6 is $F$7. For $F$8, the bound tightens dramatically to:

$F$9

• For $F$0, the threshold is $F$1 for $F$2, and for $F$3:

$F$4

This is proved via induction and a dichotomous case analysis, ruling out high-connectivity triangulations as $F$5-free for sufficiently large $F$6, and building $F$7-free planar graphs by carefully patching together small components and extremal structures.

These results showcase a series of sharp transitions: the maximal planar bound only holds for small $F$8, while for larger $F$9, increasingly sparse (yet still edge-rich) constructions become extremal.

Methodological Innovations

The authors generalize and extend the "triangular block" decomposition previously used in planar extremal graph theory. By cataloguing all possible triangular blocks in $\Theta_4$0-free graphs and considering the contribution of each block to the edge and face counts, tight inequalities are derived. They also introduce $\Theta_4$1 and $\Theta_4$2 operations (generalized vertex deletions with edge insertions) to facilitate inductive arguments on $\Theta_4$3-vertex triangulations while preserving induced path-freeness.

Further, the analysis employs fine-grained connectivity and separation arguments, such as the behavior of separating triangles and the propagation of induced paths across cutsets in planar graphs—techniques crucial to ruling out large induced paths in high-connectivity settings.

Sharp/Contradictory Claims

• Sharpness results: The bounds for $\Theta_4$4, $\Theta_4$5, and $\Theta_4$6 are shown to be tight for infinite families of $\Theta_4$7, with explicit constructions matching the upper bounds.
• Non-triangulability: For $\Theta_4$8-free planar graphs of order $\Theta_4$9, maximal planar triangulations are always excluded from being extremal; instead, the structure must be significantly more disconnected or have bounded-sized components.
• Threshold phenomena: There are precisely determined $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$0 values where the maximal planar edge count $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$1 ceases to be valid for various forbidden induced paths; e.g., for $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$2, the threshold is $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$3.

Theoretical and Practical Implications

The results provide the first comprehensive picture of induced Turán-type extremal parameters under planarity, a setting previously dominated by analyses of non-induced forbidden subgraphs. The characterization of extremal constructions for small paths refines our understanding of how planarity interacts with induced subgraph structure, which diverges significantly from the classical, non-planar setting.

On the theoretical side, the block decomposition methodology and the systemic analysis of separating sets point toward a more general program for planar Turán-type questions for larger and more involved forbidden structures. The explicit characterization of threshold phenomena sheds light on subtle transitions in extremal graph theory under combined structural constraints.

Practically, such extremal results inform the design of planar networks and facilitate bounds in distributed systems and geometric representations where planarity and avoidance of certain (induced) patterns are desired, e.g., avoiding certain propagation paths or forbidden motifs.

Open Problems and Future Directions

A range of open questions is raised:

• The complete characterization of graphs $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$4 for which $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$5 for all large $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$6 remains unresolved. The conjectured necessity of the sufficient conditions is left open for general $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$7.
• Forbidding longer induced paths ($$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$8 for $$_{(n,F^)} := \max\{|E(G)| : |V(G)|=n,\, G \text{ planar},\, F \nsubseteq_{\text{ind}} G\},$$9) is conjectured to admit similar dichotomies, with sharper bounds given for $F$0 in terms of a $F$1-vertex extremal triangulation and general upper bounds of the form

$F$2

• Further study of induced planar Turán numbers for cycles with chords (triangulated polygons) is posed, with explicit conjectured bounds and constructions detailed.

Methodologically, future work could systematize block decompositions for induced forbidden graphs $F$3 beyond cycles and paths, and explore the analogs of these results in higher-genus surfaces or for minor-excluded graph classes.

Conclusion

This work delivers a systematic, tightly-quantitative study of induced planar Turán numbers, providing sharp extremal values and explicit constructions for forbidden induced $F$4 and short paths. By leveraging block structural decomposition, novel inductive techniques, and careful combinatorial analysis, the paper clarifies the complex interaction between planarity and induced subgraph avoidance. These results not only strengthen our extremal understanding but also lay the groundwork for future investigations of induced structures in planar and minor-closed graph families.