A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti

TL;DR

This paper introduces a linear-time formula and polynomial-time algorithm for computing the strong rainbow connection number of odd cactus graphs, extending efficient computation to a broader class of graphs.

Contribution

It provides a novel polynomial-time method to compute $src(G)$ specifically for odd cactus graphs, expanding the classes of graphs with efficiently computable strong rainbow connection numbers.

Findings

Formula for $src(G)$ in odd cacti

Linear-time evaluation of the formula

Polynomial-time algorithm for optimal coloring

Abstract

We consider the problem of computing the strong rainbow connection number $src(G)$ for cactus graphs $G$ in which all cycles have odd length. We present a formula to calculate $src(G)$ for such odd cacti which can be evaluated in linear time, as well as an algorithm for computing the corresponding optimal strong rainbow edge coloring, with polynomial worst case run time complexity. Although computing $src(G)$ is NP-hard in general, previous work has demonstrated that it may be computed in polynomial time for certain classes of graphs, including cycles, trees and block clique graphs. This work extends the class of graphs for which $src(G)$ may be computed in polynomial time.