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

TL;DR

This paper characterizes the structure and uniqueness of strong central 2-trees with tail degrees 2 or 3, revealing precise degree constraints, uniqueness conditions, and growth bounds for different centrality levels.

Contribution

It provides a comprehensive structural characterization and uniqueness criteria for strong central 2-trees with tail degrees {2,3}, including explicit degree relations and bounds on graph counts.

Findings

Fan graph is unique for unicentral case.

Number of degree-3 vertices in bicentral case is always even.

Quadratic lower bound on non-isomorphic graphs for tricentral case.

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 $\Delta$ 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 $\Delta$, 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.