Edge Vertex Connectivity

When volumes of hemolymph are low, the space of the hemocoel is confined to regions where there is moisture, so we encounter problems of edge vertex connectivity. Consider the hemocoel graph having an optimal throughput from S to T. What is the smallest subset of vertices or edges, that if severed, disconnect the hemo-coel, or if we ask the question differently, what is the smallest subset of vertices or edges, that if they become non-functional, might isolate S from T?

Vertex connectivity may not be less than the combined connectivity of all edges, because if we delete one vertex incident on each edge of a cut, the graph disconnects, preventing flow of substances through it. Of course, smaller subsets of vertices might also work.

A minimum vertex degree places a lower bound on the connectivity of vertices and edges, because deleting all of a single vertex's neighbors (or the edges to all its neighboring vertices) disconnects the graph into one larger and one single-vertex portion. We may formulate many practical problems of linear programming as network flow problems indicating the power these models have. We can create special purpose network flow algorithms to solve such problems faster than we might with methods of general-purpose linear programming (Ref: Network Flow Theory).

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