Summary: Graph Partitions with Minimum Degree
Esther M. Arkin \Lambda Refael Hassin y
July 7, 1997
Given a graph with n nodes and minimum degree ffi, we give a polynomial time
algorithm that constructs a partition of the nodes of the graph into two sets X and
Y such that the sum of the minimum degrees in X and in Y is at least ffi and the
cardinalities of X and Y differ by at most ffi (ffi + 1 if n 6=
ffi (mod2)). The existence of
such a partition was shown by .
Let G = (V; E) be a simple graph of minimum degree ffi = ffi(G), with n nodes and m edges.
For S ` V define q(S) = j(i; j) 2 E : i; j 2 Sj. We define ffi(S) to be the minimum degree
in the subgraph of G induced by the nodes S. We denote by ffi(v) the degree of node v in
the graph G, and by ffi(v; S) the number of edges from v to nodes of S.
We consider the following version of a theorem of Sheehan , and give a constructive
proof. Our proof is a modification of Sheehan's proof but it leads to a polynomial algorithm
for computing the partition.
Theorem 1 There exists a partition (X; Y ) of V such that j jY j \Gamma jXj j Ÿ \Delta, ffi(X)+ ffi(Y ) –