Arboricity and Simplicial Geometric Category of Wedges and Joins of Graphs
We investigate the behavior of arboricity under two fundamental graph operations, namely wedges and joins, proving an exact formula for wedges and establishing general upper and lower bounds for joins. Using the characterization of the simplicial geometric category of connected graphs in terms of arboricity, we derive a wedge formula for simplicial geometric category and obtain corresponding estimates for graph joins. Finally, we illustrate these results through explicit computations for several classes of graphs by constructing forest decompositions and the associated covers by strongly collapsible subcomplexes.
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