# The signed matchings in graphs

• Volume: 28, Issue: 3, page 477-486
• ISSN: 2083-5892

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## Abstract

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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 ${\sum }_{e\in {E}_{G}\left(v\right)}x\left(e\right)\le 1$ for every v ∈ V(G), where ${E}_{G}\left(v\right)=uv\in E\left(G\right)|u\in V\left(G\right)$. The maximum of the values of ${\sum }_{e\in E\left(G\right)}x\left(e\right)$, 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.

## How to cite

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Changping Wang. "The signed matchings in graphs." Discussiones Mathematicae Graph Theory 28.3 (2008): 477-486. <http://eudml.org/doc/270160>.

@article{ChangpingWang2008,
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 $∑_\{e ∈ E_G(v)\} x(e) ≤ 1$ 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.},
author = {Changping Wang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {signed matching; signed matching number; maximum signed matching; signed edge cover; signed edge cover number; strongly polynomial-time},
language = {eng},
number = {3},
pages = {477-486},
title = {The signed matchings in graphs},
url = {http://eudml.org/doc/270160},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Changping Wang
TI - The signed matchings in graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2008
VL - 28
IS - 3
SP - 477
EP - 486
AB - 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.
LA - eng
KW - signed matching; signed matching number; maximum signed matching; signed edge cover; signed edge cover number; strongly polynomial-time
UR - http://eudml.org/doc/270160
ER -

## References

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10. [10] C. Wang, The signed star domination numbers of the Cartesian product graphs, Discrete Appl. Math. 155 (2007) 1497-1505, doi: 10.1016/j.dam.2007.04.008. Zbl1119.05085
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13. [13] B. Xu, Two classes of edge domination in graphs, Discrete Appl. Math. 154 (2006) 1541-1546, doi: 10.1016/j.dam.2005.12.007. Zbl1091.05055

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