On Resource Bipartitioning Problem

Given a connected graph G = (V, E), two distinct base vertices u ₁, u₂ isin V, a set V_r sube V of r special vertices and two natural numbers r₁, r₂ such that r₁ + r₂ = r, we wish to find a partition V₁, V₂ of the vertex set V such that u₁ isin V₁, u₂ isin V₂, V_i induces a connected subgraph G_i of G for each i, 1 les i les 2, V₁ contains r₂ vertices from V_r and V₂ contains r₂ vertices from V _r. We call a vertex in V_r a resource vertex and call the above problem of finding a partition as the resource bipartitioning problem. This paper gives a simple linear-time algorithm to find such a bi-partition of "path-reducible graphs". This paper also gives an algorithm for finding a resource bipartition of a connected graph G where all resource vertices are contained in the same biconnected component of G. The paper also shows that a well-known class of graphs namely "series-parallel graphs" admits a resource bipartitioning. The algorithm is based on st-numbering of G