Matroidal connectivity and conditional matroidal connectivity of star graphs

Abstract

As modern parallel computer systems continue to grow in scale, the occurrence of failures becomes inevitable, which places severe reliability challenges on the systems' underlying interconnection network. Recently, two novel indicators, named the matroidal connectivity and conditional matroidal connectivity, were proposed to better evaluate the reliability of interconnection networks. These indicators enable flexible restrictions on faulty edges in each dimension and have proven to be promising in improving the edge fault-tolerance of interconnection networks. In this paper, we aim to apply these indicators to the n-dimensional star graph Sn, which is a famous interconnection network with high symmetry and recursive hierarchical structure. We partition the faulty edge set F⊆E(Sn) into n−1 parts according to each faulty edge's dimension. By sorting the cardinality of these parts and exerting the restriction that the (n−i)-th largest part's cardinality does not exceed i!−1 for 1≤i≤n−1, we prove that Sn−F is always connected where |F|≤∑i=1n−1(i!−1). Then we determine the exact values of matroidal connectivity and conditional matroidal connectivity of star graphs. Moreover, we make comparisons to demonstrate our results outperform other similar works in terms of edge fault-tolerance.