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

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.