XMST: An Extended Minimum Spanning Tree Framework with Objective Fracture-Scale Selection

Abstract

We present XMST, an extended minimum spanning tree framework for identifying spatially coherent stellar structures using an objective, data-driven fracture-scale criterion. XMST combines percolation analysis with Jenks Natural Breaks optimisation, restricting the MST edge-length distribution to the subcritical regime before determining the final fracture scale. The method was validated using 1000 controlled Monte Carlo realisations containing eight empirical stellar-association templates, including a deliberately overlapping pair. The Percolation-Jenks fracture scale was highly reproducible, with a median of 17.028 pc and a between-realisation standard deviation of 0.066 pc. The injected structures were recovered with a mean completeness of 0.9996 and mean purity of 0.7413, while alternative CDF and Mean Edge criteria selected substantially larger fracture scales and produced markedly lower purities. Propagation of the published distance uncertainties through 10,000 additional XMST reconstructions produced a median absolute fracture-scale change of only 0.282 pc. Under these perturbations, mean completeness was 0.8605 and mean purity 0.7126, while 99.30 per cent of injected-template cases continued to satisfy the adopted detection criterion. Individual membership was less stable, with peripheral and locally sparse members showing lower persistence than structural cores. The deliberately overlapping pair was resolved spatially in only 15 of 1000 realisations; a reddening-based post-processing diagnostic increased this to 446, although with a corresponding completeness-purity trade-off. XMST therefore provides a robust method for identifying candidate stellar structures and their stable cores, while detailed outer membership and strongly overlapping populations require additional astrophysical information.