All Square Roots of a König-Egerváry Graph
Vadim E. Levit & Eugen Mandrescu
Ariel University, Israel
&
Holon Institute of Technology, Israel
June 16-19, 2015
Algorithmic Graph Theory on the Adriatic Coast 2015
Koper, Slovenia
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Outline
1
Some de…nitions : independent sets, matchings
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Outline
1
Some de…nitions : independent sets, matchings
2
König-Egerváry graphs
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Outline
1
Some de…nitions : independent sets, matchings
2
König-Egerváry graphs
3
Square of graphs
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Outline
1
Some de…nitions : independent sets, matchings
2
König-Egerváry graphs
3
Square of graphs
4
Square-roots of graphs
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Outline
1
Some de…nitions : independent sets, matchings
2
König-Egerváry graphs
3
Square of graphs
4
Square-roots of graphs
5
Some old results
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Outline
1
Some de…nitions : independent sets, matchings
2
König-Egerváry graphs
3
Square of graphs
4
Square-roots of graphs
5
Some old results
6
Our …ndings ...
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Outline
1
Some de…nitions : independent sets, matchings
2
König-Egerváry graphs
3
Square of graphs
4
Square-roots of graphs
5
Some old results
6
Our …ndings ...
7
Some open problems
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Some de…nitions: independent sets
w
G
a
wb
w
u
w
c
x w
@
w
v
@
@w
y
Figure: G has α(G ) = jfa, b, c, y gj = 4.
De…nition
An independent or a stable set is a set of pairwise non-adjacent vertices.
The independence number or the stability number α(G ) of G is
the maximum cardinality of an independent set in G .
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Some de…nitions: independent sets
w
G
a
wb
w
u
w
c
x w
@
w
v
@
@w
y
Figure: G has α(G ) = jfa, b, c, y gj = 4.
De…nition
An independent or a stable set is a set of pairwise non-adjacent vertices.
The independence number or the stability number α(G ) of G is
the maximum cardinality of an independent set in G .
Example
fag, fa, b g, fa, b, x g, fa, b, c, y g are independent sets of G .
fa, b, c, x g, fa, b, c, y g are maximum independent sets, hence α(G ) = 4.
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Some de…nitions: matchings and matching number
De…nition
A matching in G is a set of non-incident edges.
The matching number µ(G ) of G is the maximum size of a matching in G .
A matching covering all the vertices is called perfect.
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Some de…nitions: matchings and matching number
De…nition
A matching in G is a set of non-incident edges.
The matching number µ(G ) of G is the maximum size of a matching in G .
A matching covering all the vertices is called perfect.
Example
fa1 a2 g is a maximum matching in K3 , hence µ(K3 ) = 1
fv1 v2 , v3 v4 g is maximum matching in C5 , hence µ(C5 ) = 2
ft1 t2 , t3 t4 , t5 t5 g is maximum matching in G , hence µ(G ) = 3
K3
a1 w
wa2
wa3
C5
v2 w
v1 w
wv3
@
v5 @
w @wv4
G
t2 w
@
wt4
t
t1 w @w3
@
Figure: Only G has perfect matchings; e.g., M = ft1 t3 , t2 t4 , t5 t6 g.
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wt5
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Some de…nitions: König-Egerváry graphs
Remark
bjV j /2c + 1
α (G ) + µ (G )
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jV j hold for every graph G = (V , E ).
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Some de…nitions: König-Egerváry graphs
Remark
bjV j /2c + 1
α (G ) + µ (G )
jV j hold for every graph G = (V , E ).
De…nition (R. W. Deming (1979), F. Sterboul (1979))
G = (V , E ) is a König–Egerváry graph if α(G ) + µ(G ) = jV j.
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Some de…nitions: König-Egerváry graphs
Remark
bjV j /2c + 1
α (G ) + µ (G )
jV j hold for every graph G = (V , E ).
De…nition (R. W. Deming (1979), F. Sterboul (1979))
G = (V , E ) is a König–Egerváry graph if α(G ) + µ(G ) = jV j.
G1
w
a
b w
w
u
c
w
x w
@
@
w @w
v
G2
y
wv2
w
v1
wv4
@
@
w @w
v3
v5
Figure: G1 is a König–Egerváry graph, since α(G1 ) + µ(G1 ) = 7 = jV (G1 )j,
while G2 is not a König–Egerváry graph, as α(G2 ) + µ(G2 ) = 4 < 5 = jV (G2 )j.
Theorem (D. König (1931), E. Egerváry (1931))
Each bipartite graph G = (V , E ) satis…es α(G ) + µ(G ) = jV j.
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A characterization for König-Egerváry graphs
Notation
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
If S 2 Ind (G ) and H = G S, we write G = S H.
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A characterization for König-Egerváry graphs
Notation
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
If S 2 Ind (G ) and H = G S, we write G = S H.
Theorem (Levit and Mandrescu, Discrete Math. 2003)
For a graph G = (V , E ), the following properties are equivalent:
(i) G is a König-Egerváry graph;
(ii) G = S H, where S 2 Ω(G ) and jS j µ(G ) = jV S j;
(iii) G = S H, where S is an independent set with jS j jV S j and
(S, V S ) contains a matching of size jV S j.
S = fx, y g 2 Ω(G )
µ (G ) = 2 < jV
Sj
G
x w
c w
a w
wy
wb
H
f w g w
@
@
v@w
wh
wu
Figure: By above theorem, part (ii), only H is a König-Egerváry graph.
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Another characterization of König-Egerváry graphs
De…nition
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
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Another characterization of König-Egerváry graphs
De…nition
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
Theorem (Levit and Mandrescu, Discrete Applied Math. 2013)
For a graph G = (V , E ), the following properties are equivalent:
(i) G is a König-Egerváry graph;
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Another characterization of König-Egerváry graphs
De…nition
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
Theorem (Levit and Mandrescu, Discrete Applied Math. 2013)
For a graph G = (V , E ), the following properties are equivalent:
(i) G is a König-Egerváry graph;
(ii) each maximum matching is contained in (S , V S )
for some maximum independent set S ;
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Another characterization of König-Egerváry graphs
De…nition
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
Theorem (Levit and Mandrescu, Discrete Applied Math. 2013)
For a graph G = (V , E ), the following properties are equivalent:
(i) G is a König-Egerváry graph;
(ii) each maximum matching is contained in (S , V S )
for some maximum independent set S ;
(iii) each maximum matching is contained in (S, V S )
for every maximum independent set S.
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Another characterization of König-Egerváry graphs
De…nition
If A \ B = ∅ in G = (V , E ), then (A, B ) = fab 2 E : a 2 A, b 2 B g.
Theorem (Levit and Mandrescu, Discrete Applied Math. 2013)
For a graph G = (V , E ), the following properties are equivalent:
(i) G is a König-Egerváry graph;
(ii) each maximum matching is contained in (S , V S )
for some maximum independent set S ;
(iii) each maximum matching is contained in (S, V S )
for every maximum independent set S.
S = fx, y g
M = fac, yb g
G
a w
x w
wc
wy
wb
H
f w g w
@
@
v@w
Figure: M * (S, V (G ) S ), hence G is not a König-Egerváry graph.
H is a König-Egerváry graph.
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wu
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w
w
A
A
&
A
A
A
Aw
V S
S
G
'
wr r r w
w
wr r r w
w
A König-Egerváry graph G = S
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w
A
A
A
A
w
w
A
A
A
A
Aw
w
A
A
Aw
$
%
H
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Corona of graphs
De…nition
The corona of the graphs X and fHi : 1
G =X
i
n g is the graph
fH1 , H2 , ..., Hn g obtained by joining each vi 2 V (X ) to
all the vertices of Hi , where V (X ) = fvi : 1
If every Hi = H, we write G1 = X H.
X
w
G =H
v1
G1
w
v2
w
w
v1
n g.
i
K1 is a König-Egerváry graph with a perfect matching.
w
v3
w
w
v2
w
v4
w
w
v3
w
w
v4
Figure: The graphs G1 = X
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w
w
A
K3 A
wA
@ A
@A
G2
Aw
@
v1
K1 and G2 = X
Square Roots
K2
w
w
v2
w
w
w
w
w
v3
w
P3
wK1
v4
fK3 , K2 , P3 , K1 g.
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Square of a graph
De…nition
The square of the graph G = (V , E ) is the graph G 2 = (V , U ),
where xy 2 U if and only if x 6= y and distG (x, y ) 2.
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Square of a graph
De…nition
The square of the graph G = (V , E ) is the graph G 2 = (V , U ),
where xy 2 U if and only if x 6= y and distG (x, y ) 2.
w
w
C4
w
w
K1,3
w
w
w
w
K3 + e
w
@
w
w
@
@w
K4
Figure: Non-isomorphic graphs having the same square.
w
w
@
@
w @w
Example
2 = (K + e )2 = K 2 = K .
C42 = K1,3
4
3
4
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Square of a graph
De…nition
The square of the graph G = (V , E ) is the graph G 2 = (V , U ),
where xy 2 U if and only if x 6= y and distG (x, y ) 2.
w
w
C4
w
w
K1,3
w
w
w
w
w
@
w
K3 + e
w
@
@w
K4
Figure: Non-isomorphic graphs having the same square.
w
w
@
@
w @w
Example
2 = (K + e )2 = K 2 = K .
C42 = K1,3
4
3
4
Remark
(i) There is no G such that G 2 = C4 .
(ii) If one of the n vertices of G has n
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1 neighbors, then G 2 = Kn .
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Square root of a graph
De…nition
If there is some graph H such that H 2 = G ,
then H is called a square root of G , i.e., H 2
p
G.
A graph may have more than one square root.
Example
Every H of order n that has a vertex of degree n
1 is a root of Kn .
There are graphs having no square root.
Example
P3 has no square root, i.e., the equation H 2 = P3 has no solution.
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Some old results
Theorem (A. Mukhopadhyay, J. Combin. Th., 1967)
A connected graph G on n vertices v1 , v2 , ..., vn , has a square root if and
only if there exists an edge clique cover Q1 , ..., Qn of G such that, for all
i, j 2 f1, ..., n g, the following hold:
(i) Qi contains vi , for all i 2 f1, ..., n g; and
(ii) for all i, j 2 f1, ..., n g, Qi contains vj i¤ Qj contains vi .
v1 w
w v2
Q1 = fv1 , v2 g
Q2 = fv2 , v3 g
Q3 = fv3 , v4 g
Q4 = fv4 , v1 g
w v3
v4 w
fQ1 , Q2 , Q3 , Q4 g is the only edge clique cover of C4
C4
Figure: C4 has no square root: v3 2 Q2 , while v2 2
/ Q3 .
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Some old results
Theorem (A. Mukhopadhyay, J. Combin. Th., 1967)
A connected graph G on n vertices v1 , v2 , ..., vn , has a square root if and
only if there exists an edge clique cover Q1 , ..., Qn of G such that, for all
i, j 2 f1, ..., n g, the following hold:
(i) Qi contains vi , for all i 2 f1, ..., n g; and
(ii) for all i, j 2 f1, ..., n g, Qi contains vj i¤ Qj contains vi .
Example
The edge clique cover Q1 , Q2 , Q3 , Q4 satis…es (i) and (ii).
G
v1 w
@
w v2
v4 w @w v3
@
Levit & Mandrescu (AU & HIT)
Q1 = fv1 , v3 g
Q2 = fv2 , v4 g
Q3 = fv1 , v3 , v4 g
Q4 = fv2 , v3 , v4 g
P4 is a square root of G
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Some algorithmic results
Theorem (D.J. Ross and F. Harary, Bell System Tech. J., 1960)
Tree roots of a graph, when they exist, are unique up to isomorphism.
Theorem (Y. L. Lin, S. Skiena, LNCS 557, 1991)
There is an O (m ) time algorithm for …nding the square roots of a planar
graph.
Theorem (Y. L. Lin, S. Skiena, LNCS 557, 1991)
The tree square root of a graph can be found in O (m ) time, where m
denotes the number of edges of the given tree square root.
Theorem (Y. L. Lin, S. Skiena, SIAM J. of Discrete Math, 1995)
There is a linear time algorithm to recognize squares G 2 of graphs, where
G is a tree.
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Problem (SqR)
A Square Root of a Graph
Instance: A graph G .
Question: Does there exist a graph H such that H 2 = G ?
Theorem (R. Motwani, M. Sudan, Discrete Applied Math, 1994)
Problem SqR is NP-complete.
Theorem (L.C. Lau, D.G. Corneil, SIAM J. Discrete Math, 2004)
The Problem SqR remains NP-complete for chordal graphs.
Theorem (Martin Milaniµc, Oliver Schaudt, Discrete Applied Math,
2013)
The Problem SqR is polynomial for trivially perfect and threshold graphs.
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Problem (SqR)
A Square Root of a Graph
Instance: A graph G .
Question: Does there exist a graph H such that H 2 = G ?
Theorem (R. Motwani, M. Sudan, Discrete Applied Math, 1994)
Problem SqR is NP-complete.
Theorem (L.C. Lau, D.G. Corneil, SIAM J. Discrete Math, 2004)
The Problem SqR remains NP-complete for chordal graphs.
A chordal graph is one whose cycles on q
G1
w
w
w
w
w
w
G2
4 vertices have a chord.
w
w
w
H
@HH@
@ H@
@w
w @w HH
G3
w
w
Figure: Chordal graphs: only G2 has square roots, namely G1 2
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p
w
w
@
@
w @w
G2 .
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Problem (SqR)
A Square Root of a Graph
Instance: A graph G .
Question: Does there exist a graph H such that H 2 = G ?
Theorem (Martin Milaniµc, Oliver Schaudt, Discrete Applied Math,
2013)
The Problem SqR is polynomial for trivially perfect graphs.
G is a trivially perfect graph if each of its induced subgraphs H has
α(H ) maximal cliques (M. C. Golumbic, Discrete Math. 1978).
They are exactly the (P4 and C4 )-free graphs (Golumbic, DM 1978).
G1
w
w
w
w
w
w
G2
w
w
w
H
@H @
@HH@
Hw
w @w H@
G3
Figure: Only G2 , G3 are Square
trivially
perfect, and G1 2
Roots
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w
p
G2 .
w
@
w
w
@
@w
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
w
1
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
w
1
w
2
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
w
1
w
2
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w
3
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
w
1
w
2
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w
4
w
3
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
w
1
w
2
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w
4
w
3
Square Roots
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5
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Threshold graphs
De…nition
A graph G = (V , E ) is called threshold (V. Chvatal and P. L. Hammer,
1977) if it can be obtained from K1 by iterating, in any order, the
operations of adding a new vertex which is connected to
no other vertex (i.e., an isolated vertex) or
every other vertex (i.e., a dominating vertex).
Example
K1,n and Kn are threshold graphs.
G
w
2
w
w
@
@
w @w
1
6
w
3
4
5
Figure: G is a threshold graph : 4 and 6 are dominating vertices.
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Problem (SqR)
A Square Root of a Graph
Instance: A graph G .
Question: Does there exist a graph H such that H 2 = G ?
Theorem (Martin Milaniµc, Oliver Schaudt, Discrete Applied Math,
2013)
The Problem SqR is polynomial for threshold graphs.
Threshold graphs are exactly the (P4 and C4 and 2K2 )-free graphs
(V. Chvatal, P. L. Hammer, 1977).
Examples
G1 and G2 are threshold graphs, but only G2 has square roots.
v1 w
wv5
wv6
@
@
G1
@
@
v2 w @wv3 @wv4
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v1 w
wv4
@
G2
@
v2 w @wv3
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In what follows, we discuss:
1
Which König-Egerváry graphs have square roots?
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In what follows, we discuss:
1
Which König-Egerváry graphs have square roots?
2
How to compute a square root of a König-Egerváry graph?
Levit & Mandrescu (AU & HIT)
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In what follows, we discuss:
1
Which König-Egerváry graphs have square roots?
2
How to compute a square root of a König-Egerváry graph?
3
How to compute all square roots of a König-Egerváry graph?
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In what follows, we discuss:
1
Which König-Egerváry graphs have square roots?
2
How to compute a square root of a König-Egerváry graph?
3
How to compute all square roots of a König-Egerváry graph?
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In what follows, we discuss:
1
Which König-Egerváry graphs have square roots?
2
How to compute a square root of a König-Egerváry graph?
3
How to compute all square roots of a König-Egerváry graph?
Example
The graph G2 has G1 as a square root, i.e., G1 2
G3 has no square roots, because it has a leaf.
G1
w
w
w
w
w
w
G2
w
w
w
H
@H @
@HH@
Hw
w @w H@
G3
p
G2 .
w
w
w
w
@
@
w @w
Figure: König-Egerváry graphs.
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
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Theorem
If a connected König-Egerváry graph G of order
3 has a square root,
then G has perfect matchings and a unique maximum independent set.
Example
The graph G2 has G1 as a square root.
G3 has no square roots, because it has a leaf.
G1
w
w
w
w
w
w
G2
w
w
w
H
@HH@
@ H@
Hw
w @w H@
G3
w
w
w
@
w
w
@
@w
Figure: König-Egerváry graphs.
The converse of theorem above is not necessarily true; e.g., G3 .
Levit & Mandrescu (AU & HIT)
Square Roots
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Squares, roots and König-Egerváry graphs
Theorem (L & M, Graphs and Combinatorics, 2013)
For a graph H of order n
(i)
H2
2, the following are equivalent:
is a König-Egerváry graph;
(ii) H has a perfect matching consisting of pendant edges.
Corollary
Each square root of a König-Egerváry graph G , if any,
is of the form H0
H
w
w
w
w
K1 for some graph H0 .
w
w H0
H2
w
w
w
H
@HH@
@ H@
Hw
w @w H@
Figure: König-Egerváry graphs: H = H0
Levit & Mandrescu (AU & HIT)
Square Roots
K1 and H 2 .
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Squares, roots and König-Egerváry graphs
There are König-Egerváry graphs, whose squares are not
König-Egerváry graphs. E.g., every C2n .
There are non-König-Egerváry graphs, whose squares are not
König-Egerváry graphs. E.g., every C2n +1 .
Theorem (L & M, Graphs and Combinatorics, 2013)
For a graph H of order n
(i)
H2
2, the following are equivalent:
is a König-Egerváry graph;
(ii) H has a perfect matching consisting of pendant edges.
Corollary
Each square root of a König-Egerváry graph G , if any,
is of the form H0
Levit & Mandrescu (AU & HIT)
K1 for some graph H0 .
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Simplicial graphs
De…nition (G. H. Cheston, E. O. Hare, S. T. Hedetniemi and R. C.
Laskar, Congressus Numer 67, 1988)
A vertex v is simplicial in G if NG (x ) is a clique. A simplex is a clique
containing at least one simplicial vertex. G is a simplicial graph if each of
its vertices is either simplicial or adjacent to a simplicial vertex.
Theorem (Cheston et al., Congressus Numer 67, 1988)
If G is a simplicial graph and Q1 , ..., Qq are the simplices of G , then
V (G ) = [fV (Qi ) : 1 i q g and q = α(G ).
H
w
w
w
w
w
w
w
w
w
wH0
G
w
w
w
w
w S0
H
@HH@
@
@
@ H@
@
@
Hw @w @w
w @w H@
Figure: König-Egerváry graphs: H = H0
Levit & Mandrescu (AU & HIT)
Square Roots
K1 and G = H 2 .
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Square root of a König-Egerváry graph
A vertex v is simplicial in G if its neighborhood NG (v ) is a clique.
G is simplicial if each of its vertices is either simplicial or adjacent to
a simplicial vertex.
Theorem
If a König-Egerváry graph G , of order n
3, has a square root, then
every vertex of its unique maximum independent set, say S0 , is simplicial.
Moreover, fNG (x ) : x 2 S0 g is an edge clique cover of G [V (G )
H
w
w
w
w
w
w
w
w
w
wH0
G
w
w
w
w
w S0
H
@
@
@HH@
@ H@
@
@
Hw @w @w
w @w H@
Figure: König-Egerváry graphs: H = H0
Levit & Mandrescu (AU & HIT)
Square Roots
S0 ] .
K1 and G = H 2 .
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Square roots of a König-Egerváry graph
Corollary
Each square root of a König-Egerváry graph G , if any,
is of the form H0
K1 for some graph H0 .
Theorem
If a König-Egerváry graph G , of order n
3, has a square root, then
every vertex of its unique maximum independent set, say S0 , is simplicial.
Moreover, fNG (x ) : x 2 S0 g is an edge clique cover of G [V (G )
H
w
w
w
w
w
w
w
w
w
wH0
G
w
w
w
w
w S0
H
@
@
@HH@
@ H@
@
@
Hw @w @w
w @w H@
Figure: König-Egerváry graphs: H = H0
Levit & Mandrescu (AU & HIT)
Square Roots
S0 ] .
K1 and G = H 2 .
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Problem (AllSqR)
All Square Roots of a König-Egerváry Graph
Instance: A connected König-Egerváry graph G .
Output: All graphs H such that H 2 = G .
Theorem
Problem AllSqR is solvable in
O
jE j jV j + jV j2 + ((∆(G ) + M (n)) jV j per (G ))
time, where per (G ) is the number of perfect matchings of G = (V , E ),
and M (n ) is the time complexity of a matrix multiplication for two n n
matrices.
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Core of a graph
De…nition (Levit and Mandrescu, Discrete Applied Math, 2002)
core(G ) is the intersection of all maximum independent sets of G .
The problem of whether core(G ) 6= ∅ is NP-complete
(Endre Boros, M. C. Golumbic, V. E. Levit, Discrete Applied Math,
2002).
Fact
G has a unique maximum independent set if and only if
core(G ) is a maximum independent set.
G
b w
w
@
x
@
@
@
a w @w @w
c
Levit & Mandrescu (AU & HIT)
w
y
d
Square Roots
w
u
w
e
wv
wf
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Checking whether a K-E graph may have a square root
Testing whether a graph has a unique maximum independent set is
NP-hard (A. Pelc, IEEE Transactions on computers, 1991).
We need to check whether a König-Egerváry graph with perfect
matchings has a unique maximum indep set, and if positive, to …nd
it.
Lemma
Let G = (V , E ) be a König-Egerváry graph having a perfect matching,
and v 2 V . Then the following assertions are true:
(i) v 2 core(G ) i¤ G v is not a König-Egerváry graph;
(ii) one can …nd core(G ) in O (jV j jE j + jV j2 ) time;
(iii) one can check whether G has a unique maximum independent
set (namely core(G )), and …nd it, in O (jV j jE j + jV j2 ) time.
Levit & Mandrescu (AU & HIT)
Square Roots
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A sketch of an algorithm generating all square roots of a
K-E graph
There is a poly time algorithm …nding a maximum matching M in G
p
that needs O (jE j
jV j) time (V. V. Vazirani, Combinatorica 1994).
If 2 jM j 6= jV j, i.e., M is not perfect, then G has no square root.
Assume that M is a perfect matching. Hence α (G ) = µ (G ) = jM j.
Since G is a König-Egerváry graph with a perfect matching,
one can …nd S0 = core(G ) in time O (jV j jE j + jV j2 ).
If α (G ) 6= jS0 j, then G has no square root, since it has more than
one maximum independent set.
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Otherwise, we infer that Ω(G ) = fS0 g and G = S0
H1 .
' $
' $' $
' $
w
a H
b w
HH
w
c H
d w
HH
wu1
H
HHw
u2
a w
H
HH
H
c w
H
HH
H
b w
wu3
d w
H
HHw
u4
wu1
HHw
u2
wu3
HHw
u4
& %
& %& %
& %
S0
G
H1
HB
One can run an algorithm generating all perfect matchings in the
bipartite graph HB = (S0 , V (H1 ) , E E (H1 )) with the time
p
complexity O
jV j jE (HB )j + per (HB ) log jV j
(T. Uno, LNCS 2223, 2001).
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In other words, every solution of the equation G = H 2 is based on a
choice of a perfect matching of the bipartite graph HB .
Let M0 = fxi yi : 1
where S0 = fxi : 1
H1
x1 w
H2
w
x1 H
jV j /2g be such a perfect matching of HB ,
jV j /2g.
wx3
wx4
G
wx2 wx3
HH
H
wy2HHwy1
y3 w
wy4
wx4
wy4
H22
y1 w
wx2
i
i
wy2
wy3
w
w
w
w
H
@H @
@
@HH@
@
H w @w
w @w H@
w
w
w
w
P
H
@P
HP
@
P
H P
PP
@ H@
Hw PPw
w @w H@
Figure: H1 , H2 are candidates for the equation H 2 = G , corresponding to
di¤erent perfect matchings of HB , but only H12 = G .
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To de…ne the edge set of the graph H as a function of the perfect
matching M0 , we proceed as follows:
keep M0 be such as a part of E (H );
check that for every xk z 2 E (G ) fxk yk g , 1 k jV j /2, there
exists the edge yk z 2 E (G ), otherwise M0 may not generate a square
root of G ;
build the graph H0 as follows:
V (H0 ) = V
S0 ,
E (H0 ) = fyk z : xk z 2 E (G )
fxk yk g , 1
k
jV j /2g ;
if (V , E (H0 ) [ M0 )2 = G , then the graph (V , E (H0 ) [ M0 ) is a
square root of G , otherwise M0 does not generate a square root.
Levit & Mandrescu (AU & HIT)
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Since S0 is the unique maximum independent set of a König-Egerváry
graph G , and, on the other hand, by a Theorem characterizing
König-Egerváry graphs, every matching of G is contained in
(S0 , V S0 ), one may conclude that the graphs G = S0 H1 and
HB = (S0 , V (H1 ) , E E (H1 )) have the same perfect matchings.
In summary, testing allpthe perfect matchings of the bipartite graph
HB one can generate G with
O
jE j jV j + jV j2 + ((∆(G ) + M (n)) jV j per (G ))
time complexity.
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Symmetric bipartite graphs
De…nition (N. Kakimura, 2008)
A bipartite graph G = (A, B, E ) with jAj = jB j is said to be
symmetric if aj bi 2 E holds for every ai bj 2 E .
Example
G1 is bipartite and symmetric, while
G2 is bipartite, but not symmetric.
w
w
w
w
w
HH PPPH
@
@
PH
H
H
PPH@
H
@
PP
P
H
@w
w
w HHw @w H
b1
G1
a1
b2
a2
b3
a3
b4
a4
b5
G2
a5
w
w
w
w
w
HH PPPH
@
@
PH
H
H
PPH@
H
@
PP
Hw
@
P
w
w HHw @w H
b1
b2
b3
b4
b5
a1
a2
a3
a4
a5
Figure: Bipartite graphs on the same vertices.
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De…nition
Let F = (A, B, E ) be a bipartite graph, such that A = faj : 1 j p g
and B = fbk : 1 k q g. The adjacency matrix of F is
Adj (F ) = (xjk )p q , where xjk = 1 if aj bk 2 E , and xjk = 0, otherwise.
Example
8 90
a1 >
>
>
< >
=
a2 B
B
Adj (F ) =
@
a3 >
>
>
>
: ;
a4
F
1
1
0
0
1
1
1
1
0
1
1
1
w
@
1
0
0 C
C and M is a maximum matching.
0 A
1
0
0
1
1
w
w
w
H
@HH@
@
@ H@
@w
w @w @w HH
b1
b2
b3
b4
a1
a2
a3
a4
w
b5
Figure: "Blue matching" is a maximum matching.
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De…nition
Let M be a perfect matching of F = (A, B, E ). The permutation matrix
PM determined by M is: PM (i, j ) = 1 if and only if aj bi 2 M.
Example
0
b1
1
1
0
0
B a1
B
Adj (F ) = B
B a2
@ a3
a4
F
b2
1
1
1
1
b3
0
1
1
1
w
@
b4
0
0
1
1
1
0
1 0 0
C
B 0 0 0
C
C =) PM = B
C
@ 0 1 0
A
0 0 1
1
0
1 C
C
0 A
0
w
w
w
H
@HH@
@
@ H@
@w
w @w @w HH
b1
b2
a1
a2
b3
b4
a3
a4
Figure: M = fa1 b2 , a2 b3 , a3 b4 , a4 b2 g is a perfect matching.
Levit & Mandrescu (AU & HIT)
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De…nition
Let M be a perfect matching of F = (A, B, E ).
The permutation matrix PM determined by M is:
PM (i, j ) = 1 if and only if aj bi 2 M.
Example
0
1 1 0
a1
B
1 1 1
a
Adj (F ) = 2 B
a3 @ 0 1 1
0 1 1
a4
w
@
w
w
w
H
@HH@
@
@ H@
@w
w @w @w HH
b1
F
0
1
1 0
0
B
C
0 0
0 C
=) PM = B
@ 0 1
1 A
0 0
1
a1
b2
b3
b4
a2
a3
a4
0
0
0
1
1
0
1 C
C
0 A
0
Figure: M = fa1 b2 , a2 b3 , a3 b4 , a4 b2 g is a perfect matching.
Levit & Mandrescu (AU & HIT)
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De…nition
Let M is a perfect matching of F = (A, B, E ). The corresponding
adjacency matrix of F with respect to M is Adj (F , M ) = Adj (F ) PM .
Example
0
1
B 1
Adj (F , M ) = B
@ 0
0
w
@
0
1
1
1
10
0
1 0
B 0 0
0 C
CB
1 A@ 0 1
1
0 0
w
w
w
H
@HH@
@
@ H@
Hw
w @w @w H@
b1
F
1
1
1
1
a1
b2
a2
b3
a3
0
0
0
1
b4
=)
a4
F
1 0
0
1 0
B 1 1
1 C
C=B
0 A @ 0 1
0
0 1
0
0
1
1
1
1
1 C
C
1 A
1
w
w
w
w
H
@
@HH@
@
@ H@
Hw
w @w @w H@
b1
b3
b4
b2
a1
a2
a3
a4
Figure: "Blue matching" is a perfect matching.
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De…nition
A perfect matching M of F = (A, B, E ) is symmetric if
Adj (Fn, M ) = Adj (F ) PM ois symmetric, i.e.,
M = ai b τ ( i ) : 1 i
jAj is symmetric if aτ 1 (j ) bτ (i ) 2 E holds for
every ai bj 2 E .
Example
0
1
B 0
Adj (F , M ) = B
@ 1
0
1
1
1
0
10
0 0
1
B 0 0
1 C
CB
0 A@ 1 0
0 1
1
w
w
w
w
H
@HHHHH
@ H
H
w @w HHw HHw
b1
F
0
1
0
1
a1
b2
a2
b3
a3
1
0
0
0
b4
a4
=)
F
1 0
1
0
B 1
1 C
C=B
0 A @ 1
0
0
1
0
1
0
1
0
1 C
C
0 A
1
w
w
w
w
H
@HHHHH
@ H
H
w @w HHw HHw
b3
b1
b2
b4
a1
a2
a3
a4
Figure: "Blue matching"
is a perfect matching.
Square Roots
Levit & Mandrescu (AU & HIT)
1
1
0
1
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De…nition
Let F = (A, B, E ) be a bipartite graph, such that A = faj : 1 j p g
and B = fbk : 1 k q g. The adjacency matrix of F is
Adj (F ) = (xjk )p q , where xjk = 1 if aj bk 2 E and xjk = 0, otherwise.
De…nition
Let F = (A, B, E ) be a bipartite graph, and M be a perfect matching of
F . The corresponding adjacency matrix of F with respect to M is
Adj (F , M ) = Adj (F ) PM .
Clearly, if F = (A, B, E ) has a perfect matching M,then Adj (F , M ) has
xkk = 1, for all k 2 f1, 2, ..., jAjg.
De…nition
Let F = (A, B, E ) be a bipartite graph. A perfect matching M is
symmetric
if Adj (F , M ) is osymmetric. In other words a perfect matching
n
M = ai b τ ( i ) : 1 i
jAj is symmetric if aτ 1 (j ) bτ (i ) 2 E holds for
every ai bj 2 E .
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De…nition
n
A perfect matching M = ai bτ (i ) : 1
is symmetric if aτ
1 (j )
i
o
jAj in F = (A, B, E )
bτ (i ) 2 E holds for every ai bj 2 E .
A bipartite graph may have both symmetric and non-symmetric
perfect matchings.
Example
M1 = fai bi : 1
5g and M2 = fa1 b1 , a2 b2 , a3 b4 , a4 b5 , a5 b3 g
i
are perfect matchings, but only M1 is symmetric.
w
w
w
w
w
HH PPPH
@
H
@
P
H
PH
H
@
PH
P@
PH
P
@w
w
w HHw @w H
b1
M1
a1
b2
a2
b3
a3
b4
a4
b5
M2
a5
w
w
w
w
PP H
HH w
P
@
H
@
P
H
PH
H
@
PH
P@
P@
Hw
P
w
w HHw @w H
b1
b2
b4
b5
b3
a1
a2
a3
a4
a5
Figure: Both M1 and M2 are perfect matchings of the same bipartite graph.
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Projection with respect to a perfect matching
De…nition
The projection of F = (A, B, E ) on A with respect to a perfect matching
M = f ai b i : 1
jAjg is a graph P = P (F , M, A) de…ned as follows:
i
V (P ) = A and E (P ) = fai aj : ai bj 2 E or aj bi 2 E g.
Example
The projection P = P (F , M, A) of F = (A, B, E ) on A
with respect to the perfect matching M = fai bi : 1
w
@
w
w
w
HH @
HH@
@
Hw
w @w
w H@
b1
F
a1
b2
a2
b3
a3
Levit & Mandrescu (AU & HIT)
b4
a4
w
5g.
i
b5
w
P
a5
Square Roots
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a1
w
a2
w
a3
w
a4
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a5
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A new interpretation of an old result
Theorem (A. Mukhopadhyay, J. Combin. Th., 1967)
A connected graph G on n vertices v1 , v2 , ..., vn , has a square root if and
only if there exists an edge clique cover Q1 , ..., Qn of G such that, for all
i, j 2 f1, ..., n g, the following hold:
(i) Qi contains vi , for all i 2 f1, ..., n g; and
(ii) for all i, j 2 f1, ..., n g, Qi contains vj i¤ Qj contains vi .
I.e., the fact that G has a square root means that a natural matching
fvi Qi : 1 i ng in the vertex-clique bipartite graph is symmetric.
Q1 Q2 Q3 Q4 Q5
w
w
w
w
w
H
@H @
@
@
@HH@
@
@
Hw @w @w
w @w H@
G
v1
v2
v3
v4
v5
Figure: Vertex-clique bipartite graph.
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Square roots of a König-Egerváry graph
H
w
w
w
w
w
w
w
w
w
wH0
G
w
w
w
w
w S0
H
@H @
@
@
@HH@
@
@
Hw @w @w
w @w H@
Figure: König-Egerváry graphs: H = H0
K1 and G = H 2 .
Q1 = fv1 , v2 , v3 g , Q2 = fv1 , v2 , v3 g,
Q3 = fv1 , v2 , v3 , v4 g , Q4 = fv3 , v4 , v5 g , Q5 = fv4 , v5 g
H
w
w
w
w
w
w
w
w
w
wH0
G
w
w
w
w
w
H
@HH@
@
@
@ H@
@
@
Hw @w @w
w @w H@
Q1
Q2
Q3
v1
v2
v3
Figure: König-Egerváry graphs: H = H0
Levit & Mandrescu (AU & HIT)
Square Roots
Q4
v4
Q5
v5
K1 and G = H 2 .
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Theorem (A. Mukhopadhyay, J. Combin. Th., 1967)
A connected graph G on n vertices v1 , v2 , ..., vn , has a square root if and
only if there exists an edge clique cover Q1 , ..., Qn of G such that, for all
i, j 2 f1, ..., n g, the following hold: (i) vi 2 Qi , for all i 2 f1, ..., n g; and
(ii) for all i, j 2 f1, ..., n g, Qi contains vj i¤ Qj contains vi .
Theorem
If a König-Egerváry graph G , of order n
3, has a square root, then
every vertex of its unique maximum independent set, say S0 , is simplicial.
Moreover, fNG (x ) : x 2 S0 g is an edge clique cover of G [V (G )
G
x1
S0 w
H
w
w
w
w
@
@
@HH@
BC (G )
@ H@
@
@
Hw @w @w
w @w H@
v1
x2
v2
x3
v3
x4
v4
S0 ] .
w
w
w
w
w
H
@HH@
@
@
@ H@
@
@
Hw @w @w
w @w H@
x5
Q1
Q2
Q3
v5
v1
v2
v3
Q4
v4
Q5
v5
Figure: A König-Egerváry graph G and its vertex-clique bipartite graph BC (G ).
Levit & Mandrescu (AU & HIT)
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De…nition (Double Covering)
Let G = (V , E ) , V = fv1 , v2 , ..., vn g , and V̂ = fv̂1 , v̂2 , ..., v̂n g . The
double covering of G is the bipartite graph B (G ) with the bipartition
V , V̂ and edges vi v̂j and vj v̂i for every edge vi vj 2 E .
Theorem (R. A. Brualdi, F. Harary, Z. Miller, J. Graph Theory, 1980)
B (G ) is connected i¤ G is connected and non-bipartite.
Theorem (Dragan Marusic, R. Scapellato, N. Zagagha Salvi)
Let A be a g -matrix (a square symmetric (0, 1) matrix with the 0 (zero)
principal diagonal) of order n , and R be a permutation matrix
representing an n-cycle. Then A R is a g -matrix if and only if A = 0.
Levit & Mandrescu (AU & HIT)
Square Roots
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Theorem
Let A be a g -matrix (a square symmetric (0, 1) matrix with the 0 (zero)
principal diagonal) of order n , and R be a permutation matrix
representing an n-cycle. Then A R is a g -matrix if and only if A = 0 .
Proof.
A Latin Square Sketch of the Proof:
1
1
2
1
2 1
=)
=)
1
2 1
1
2 1
Levit & Mandrescu (AU & HIT)
1
2
3
4
Square Roots
4
1
2
3
3
4
1
2
2
3
4
1
19/06
48 / 55
Theorem
Let A be a g -matrix (a square symmetric (0, 1) matrix with the 0 (zero)
principal diagonal) of order n , and R be a permutation matrix
representing an n-cycle. Then A R is a g -matrix if and only if A = 0 .
Proof.
A Latin Square Sketch of the Proof:
1
1
2
1
2 1
=)
=)
1
2 1
1
2 1
x1
x2
x3
x4
y1 y2 y3 y4
1 1 0 1
1 1 1 0
0 1 1 1
1 0 1 1
Levit & Mandrescu (AU & HIT)
x1
x2
x3
x4
1
2
3
4
4
1
2
3
3
4
1
2
2
3
4
1
y1 y2 y4 y3
1 1 1 0
1 1 0 1
0 1 1 1
1 0 1 1
Square Roots
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Theorem
If F = (A, B, E ) has symmetric perfect matchings and twins,
then it has at least two symmetric perfect matchings.
Proof.
Let M = fai bi : 1
and bj , bk be twins.
i
q g be a symmetric perfect matching of F ,
Then the columns of the matrix Adj (F , M ) ,
corresponding to bj and bk , are identical.
Thus interchanging these two columns leaves the matrix symmetric.
Hence the principal diagonal of the new matrix de…nes another perfect
matching, that is symmetric, as well.
Levit & Mandrescu (AU & HIT)
Square Roots
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Remark
If F = (A, B, E ) has no twins, then it may have more than one symmetric
perfect matching.
Example
G = (A, B, E ) has no twins, while the perfect matchings
M1 = fai bi : 1 i 4g,
M2 = fa1 b2 , a2 b1 , a3 b4 , a4 b3 g
M3 = fa1 b3 , a2 b4 , a3 b1 , a4 b2 g are symmetric.
w
w
w
w
H
@HHHHH@
@ H
H@
Hw
w @w HHw H@
w
w
w
w
P
@PP@
PP @
@
@ PP@
@w
w @w @w PP
w
w
w
PP w
H
H
HP
@
H
P
H PH
H@ PH
PP
H
Pw
Hw H
w
w H@
b1
b2
b3
b4
b2
b1
b4
b3
b3
b4
b1
b2
a1
a2
a3
a4
a1
a2
a3
a4
a1
a2
a3
a4
Figure: A bipartite graph G = (A, B, E ) and three of its perfect matchings.
Levit & Mandrescu (AU & HIT)
Square Roots
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0
1
B 1
Perm B
@ 0
1
1
1
1
0
0
1
1
1
Levit & Mandrescu (AU & HIT)
1
1
0 C
C=9
1 A
1
Square Roots
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0
1
B 1
Perm B
@ 0
1
1
1
1
0
0
1
1
1
1
1
0 C
C=9
1 A
1
1 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
Levit & Mandrescu (AU & HIT)
Square Roots
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0
1
B 1
Perm B
@ 0
1
1
1
1
0
0
1
1
1
1
1
0 C
C=9
1 A
1
1 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
2 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
Levit & Mandrescu (AU & HIT)
2 y1 y2 y4 y3
x1 1 1 1 0
x2 1 1 0 1
x3 0 1 1 1
x4 1 0 1 1
Square Roots
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4 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
Levit & Mandrescu (AU & HIT)
4 y2 y3 y4 y1
x1 1 0 1 0
x2 1 1 0 1
x3 1 1 1 1
x4 0 1 1 1
Square Roots
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4 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
4 y2 y3 y4 y1
x1 1 0 1 0
x2 1 1 0 1
x3 1 1 1 1
x4 0 1 1 1
5 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
5 y2 y1 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 1 0 1 1
x4 0 1 1 1
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
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4 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
4 y2 y3 y4 y1
x1 1 0 1 0
x2 1 1 0 1
x3 1 1 1 1
x4 0 1 1 1
5 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
5 y2 y1 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 1 0 1 1
x4 0 1 1 1
6 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
6 y2 y1 y4 y3
x1 1 1 1 0
x2 1 1 0 1
x3 1 0 1 1
x4 0 1 1 1
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
52 / 55
7 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
Levit & Mandrescu (AU & HIT)
7 y4 y1 y2 y3
x1 1 1 1 0
x2 0 1 1 1
x3 1 0 1 1
x4 1 1 0 1
Square Roots
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7 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
8 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
Levit & Mandrescu (AU & HIT)
7 y4 y1 y2 y3
x1 1 1 1 0
x2 0 1 1 1
x3 1 0 1 1
x4 1 1 0 1
8 y4 y3 y2 y1
x1 1 0 1 1
x2 0 1 1 1
x3 1 1 1 0
x4 1 1 0 1
Square Roots
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7 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
8 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
9 y1 y2 y3 y4
x1 1 1 0 1
x2 1 1 1 0
x3 0 1 1 1
x4 1 0 1 1
Levit & Mandrescu (AU & HIT)
7 y4 y1 y2 y3
x1 1 1 1 0
x2 0 1 1 1
x3 1 0 1 1
x4 1 1 0 1
8 y4 y3 y2 y1
x1 1 0 1 1
x2 0 1 1 1
x3 1 1 1 0
x4 1 1 0 1
9 y4 y2 y3 y1
x1 1 1 0 1
x2 0 1 1 1
x3 1 1 1 0
x4 1 0 1 1
Square Roots
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7 y1
x1 1
x2 1
x3 0
x4 1
8 y1
x1 1
x2 1
x3 0
x4 1
9 y1
x1 1
x2 1
x3 0
x4 1
0
y2 y3
1 0
1 1
1 1
0 1
y2 y3
1 0
1 1
1 1
0 1
y2 y3
1 0
1 1
1 1
0 1
1 1 0
B 1 1 1
Perm B
@ 0 1 1
1 0 1
Levit & Mandrescu (AU & HIT)
y4
1
0
1
1
y4
1
0
1
1
y4
1
0
1
1
1
0
1
1
7 y4 y1 y2 y3
x1 1 1 1 0
x2 0 1 1 1
x3 1 0 1 1
x4 1 1 0 1
8 y4 y3 y2 y1
x1 1 0 1 1
x2 0 1 1 1
x3 1 1 1 0
x4 1 1 0 1
9 y4 y2 y3 y1
x1 1 1 0 1
x2 0 1 1 1
x3 1 1 1 0
x
1 0 1 1
14
0
1
C
B 1
C=9
Sym B
A
@ 0
1
Square Roots
1
1
1
0
0
1
1
1
1
1
0 C
C=3
1 A
1
19/06
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An induced matching in a graph G is a matching M where no two
edges of M are joined by an edge.
Every induced macthing in a bipartite graph is symmetric as well.
Consequently, the size of a maximum symmetric matching is
greater or equal to the size of a maximum induced matching.
The problem of …nding a maximum induced matching is NP-hard,
even for bipartite graphs (K. Cameron, Discrete Applied Math, 1989;
L. J. Stockmeyer and V. V. Vazirani, Inform. Proc. Letters, 1982).
Example
w
w
w
PP w
HH w
@
HH PPPP @
P
@w
w
w HHw
w PP
b1
G1
a1
b2
a2
b3
a3
b4
a4
b5
G2
a5
w
H
w
w
w
w
H
@H
HH
H
H
@ H
w
w HHw @w HHw
b1
b2
b3
b4
b5
a1
a2
a3
a4
a5
Figure: "Blue matchings" are induced matchings.
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
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So Much for Today, but ...
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
55 / 55
So Much for Today, but ...
Problem
Estimate the number of symmetric perfect matchings of a balanced
bipartite graph.
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
55 / 55
So Much for Today, but ...
Problem
Estimate the number of symmetric perfect matchings of a balanced
bipartite graph.
Problem
Find the size of a maximum symmetric matching of a bipartite graph.
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
55 / 55
So Much for Today, but ...
Problem
Estimate the number of symmetric perfect matchings of a balanced
bipartite graph.
Problem
Find the size of a maximum symmetric matching of a bipartite graph.
Problem
Given a balanced bipartite graph without twins and a symmetric perfect
matching, …nd another symmetric perfect macthing, if any.
Levit & Mandrescu (AU & HIT)
Square Roots
19/06
55 / 55
So Much for Today, but ...
Problem
Estimate the number of symmetric perfect matchings of a balanced
bipartite graph.
Problem
Find the size of a maximum symmetric matching of a bipartite graph.
Problem
Given a balanced bipartite graph without twins and a symmetric perfect
matching, …nd another symmetric perfect macthing, if any.
Conjecture
All square-roots of a König-Egerváry graph G are isomorphic.
Levit & Mandrescu (AU & HIT)
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