Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs
Let G be a simple graph with 2 n vertices and a perfect matching. The forcing number f ( G, M ) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G . Among all perfect matchings M of G , the minimum and maximum values of f ( G...
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Published in | Acta mathematica Sinica. English series Vol. 39; no. 7; pp. 1289 - 1304 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2023
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | Let
G
be a simple graph with 2
n
vertices and a perfect matching. The forcing number
f
(
G, M
) of a perfect matching
M
of
G
is the smallest cardinality of a subset of
M
that is contained in no other perfect matching of
G
. Among all perfect matchings
M
of
G
, the minimum and maximum values of
f
(
G, M
) are called the minimum and maximum forcing numbers of
G
, denoted by
f
(
G
) and
F
(
G
), respectively. Then
f
(
G
) ≤
F
(
G
) ≤
n
− 1. Che and Chen (2011) proposed an open problem: how to characterize the graphs
G
with
f
(
G
) =
n
− 1. Later they showed that for a bipartite graph
G, f
(
G
)=
n
− 1 if and only if
G
is complete bipartite graph
K
n,n
. In this paper, we completely solve the problem of Che and Chen, and show that
f
(
G
)=
n
− 1 if and only if
G
is a complete multipartite graph or a graph obtained from complete bipartite graph
K
n,n
by adding arbitrary edges in one partite set. For all graphs
G
with
F
(
G
) =
n
− 1, we prove that the forcing spectrum of each such graph
G
forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs
G
form an integer interval from
⌊
n
2
⌋
to
n
− 1. |
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ISSN: | 1439-8516 1439-7617 |
DOI: | 10.1007/s10114-023-1020-6 |