Strong Central 2-Trees with Tail Degrees {2, 3}: Structural Characterization and Uniqueness Criteria

TL;DR

This work develops a unified framework for strong $r$-central 2-trees with tail degrees in $\{2,3\}$, deriving universal degree-parameter constraints and then giving complete descriptions across the unicentral, bicentral, and tricentral regimes. It establishes a sharp rigidity–growth spectrum: a unique fan realization in the unicentral case, bounded uniqueness with explicit constructions and parity constraints in the bicentral case, and controlled, yet quadratic, combinatorial growth in the tricentral case with extremal all-degree-2-tail structures. The results illuminate structural transitions and yield both exact realizability results and non-isomorphic enumeration bounds, complemented by complete enumerative data for small graphs. Collectively, they provide a cohesive structural and enumerative theory for central 2-trees with bounded tail degrees and highlight sharp transitions from rigidity to combinatorial expansion.

Abstract

We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $Δ$ to the numbers of degree-$3$ and degree-$2$ tail vertices. In the unicentral case ($r=1$), we prove that the fan graph is the unique realization for all $n\ge 3$. For bicentral $2$-trees ($r=2$), we show that the number of degree-$3$ vertices is always even, establish sharp uniqueness results for $x\in\{0,2\}$, prove existence for all feasible values of $Δ$, and obtain linear lower bounds on the number of non-isomorphic realizations. For tricentral $2$-trees ($r=3$), we characterize extremal configurations, establish a divisibility constraint on the tail parameters, and prove a quadratic lower bound on the number of non-isomorphic graphs for infinitely many values of $n$. These results provide a unified structural framework for central $2$-trees with bounded tail degrees and highlight sharp transitions between rigidity and combinatorial growth.

Theorems & Definitions (47)

• Theorem 1.1: Bose et al. bose2008
• Definition 1.2: Central 2-trees
• Lemma 2.1: Tail parameters for central $2$-trees
• proof
• Corollary 2.2: Necessary range for $\Delta$
• proof
• Corollary 2.3
• Lemma 3.1: Tail parameters for $r=1$
• proof
• Theorem 3.2: Unicentral classification with tail $\{2,3\}$