An efficient algorithm for generating transmission irregular trees
Ivan Stošić, Ivan Damnjanović
TL;DR
The paper addresses the problem of generating all transmission irregular (TI) trees up to a given order more efficiently than brute-force filtration. It introduces weakly transmission irregular (WTI) trees and an unbalanced canonical representation to decompose TI trees, then builds TI trees by joining WTI components with a new root while preserving level-wise transmission distinctness. The authors present a concrete generation framework, including data structures and algorithms (generateWTITrees, generateTITrees) implemented in C++20, and demonstrate substantial speedups over existing tools, enabling TI-tree enumeration up to at least order $42$ within feasible time. This work provides a practical, scalable tool for TI-tree exploration with potential implications for transmission-based graph characterizations and related combinatorial generation problems.
Abstract
The transmission of a vertex in a connected graph is the sum of distances from that vertex to all the other vertices. A connected graph is transmission irregular if any two distinct vertices have different transmissions. We present an efficient algorithm that generates all the transmission irregular trees up to a given order, up to isomorphism.