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Discussiones Mathematicae Graph Theory 28(3) (2008) 477-486
DOI: https://doi.org/10.7151/dmgt.1421

THE SIGNED MATCHINGS IN GRAPHS

Changping Wang

Department of Mathematics
Ryerson University
Toronto, ON, Canada, M5B 2K3
e-mail: cpwang@ryerson.ca

Abstract

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 ∈ EG(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 β₁^′(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 β₁^′(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 β₁^′(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.

2000 Mathematics Subject Classification: 05C70, 05C85.

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Received 18 December 2007
Revised 8 May 2008
Accepted 28 July 2008