The minimum size of a graphic parity network for a connected graph is at least m+n-1, rising to m+Omega(n^1.5) for graphs with no short cycles, and a randomized construction achieves m+O(n^1.5 sqrt(log n)).
Tractability of parameterized com- pletion problems on chordal, strongly chordal, and proper interval graphs.SIAM Journal on Computing, 28(5):1906–1922, 1999
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Toward Minimum Graphic Parity Networks
The minimum size of a graphic parity network for a connected graph is at least m+n-1, rising to m+Omega(n^1.5) for graphs with no short cycles, and a randomized construction achieves m+O(n^1.5 sqrt(log n)).