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• Mar 24, 2025 : = beta(G), where beta(G) represents the sum of the orders of the bricks in G. To develop our techniques, we examined lambda-matchability in bipartite graphs. A straightforward counting argument reveals that no vertex in a bipartite matching covered graph is lambda-matchable. Extending this idea, an (a, b)-matching is defined as a spanning subgraph where two designated vertices, a and b, have degree three, while all other vertices have degree one. If such a subgraph exists, the pair (a, b) is deemed lambda-matchable. We also establish a lower bound on the number of lambda-matchable pairs in any bipartite 2-connected cubic graph, expressed in terms of its braces. Our approach first addresses the 3-connected case before generalizing to 2-connected graphs using 2-cuts as an induction tool. Additionally, we characterize graphs that attain these lower bounds with equality, providing insight into their structural properties."> lamda - matchability in cubic graphs - G S Santhosh Raghul (IITM)