Geodesic
The signed matchings in graphs
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 3, pp. 477-486.

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Let G be a graph with vertex set V(G) and edge set E(G). A signed matching is a function x: E(G) → -1,1 satisfying ∑_e ∈ E_G(v) x(e) ≤ 1 for every v ∈ V(G), where E_G(v) = uv ∈ E(G)| u ∈ V(G). The maximum of the values of ∑_e ∈ E(G) x(e), taken over all signed matchings x, is called the signed matching number and is denoted by β'₁(G). In this paper, we study the complexity of the maximum signed matching problem. We show that a maximum signed matching can be found in strongly polynomial-time. We present sharp upper and lower bounds on β'₁(G) for general graphs. We investigate the sum of maximum size of signed matchings and minimum size of signed 1-edge covers. We disprove the existence of an analogue of Gallai's theorem. Exact values of β'₁(G) of several classes of graphs are found.
Keywords: signed matching, signed matching number, maximum signed matching, signed edge cover, signed edge cover number, strongly polynomial-time 
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Wang, Changping. The signed matchings in graphs. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 3, pp. 477-486. http://geodesic.mathdoc.fr/item/DMGT_2008_28_3_a7/

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