Download Vertex covers

Vertex covers
Definition A vertex cover of a graph G is a subset of vertices
S ✓ V (G) such that every edge has an endpoint in S. (That is, for
all e 2 E(G), e \ S 6= ;.)
Example Vertices represent guard stations at an art gallery, while
edges represent hallways.
Find a vertex cover
(so that all hallways are guarded).
vertex cover of size 5
Definition The vertex covering number of G is the minimum
cardinality of a vertex cover and is denoted (G).
Example
Petersen graph
Remark
NP-complete decision problem
Instance: graph G, pos. integer k
Question: Is there a vertex cover
of G of size k?
(G) = 6
A useful tool: for an induced subgraph H of graph G, (G)
(H).
Math 104 - Prof. Kindred - Lecture 7
Page 1
Question
What is the relationship between vertex covers and
matchings of a graph?
Example
Petersen graph has a vertex cover of size 6 and a matching of size 5
Proposition. For any graph G, ↵0(G)  (G).
Proof. Let M be a maximum matching of G, and let S be a minimum
vertex cover. For every edge e in M , there exists an endpoint of e in
S. Thus, because edges of M share no endpoints,
↵0(G) = |M |  |S| = (G).
⌅
Corollary. If G = (V, E) is a graph, M ✓ E is a matching of
G, and S ✓ V is a vertex cover of G such that |S| = |M |, then
• M is maximum,
• S is minimum,
• and ↵0(G) = |M | = |S| = (G).
Example
↵0(G) = (G) = 4
What kind of graph is G?
Math 104 - Prof. Kindred - Lecture 7
Page 2
König-Egerváry theorem
Are there certain kinds of graphs G for which ↵0(G) = (G)?
Theorem (König-Egerváry theorem). If G is a bipartite graph,
then ↵0(G) = (G).
Proof. Let X, Y be a bipartition of bipartite graph G, and let S be
a minimum vertex cover of G.
Strategy: construct a matching of size |S|.
Let
A = X \ S,
B = Y \ S,
C = X A,
D = Y B.
A
D
C
B
X
Y
We focus on induced subgraphs G[A [ D] and G[B [ C].
Claim: For any subset T ✓ A, |NG[A[D](T )|
|T |.
Proof. Suppose, for sake of contradiction, that there exists
a subset T of A such that |NG[A[D](T )| < |T |. Swapping
T for NG[A[D](T ) in S, i.e., S T [ NG[A[D](T ), yields a
smaller vertex cover of G than S. )( This contradicts the
minimality of S.
⌅
Math 104 - Prof. Kindred - Lecture 7
Page 3
This claim verifies Hall’s condition for both G[A [ D] and G[B [ C].
Thus, by Hall’s theorem,
9 matching M 0 in G[A [ D] saturating A,
and 9 matching M 00 in G[B [ C] saturating B.
So M = M 0 [ M 00 is a matching with
|M | = |M 0| + |M 00| = |A| + |B| = |S|,
and therefore, |M | = |S|, so ↵0(G) = (G) by previous corollary.
⌅
We can prove Hall’s theorem using the König-Egerváry theorem and
vice-versa.
Math 104 - Prof. Kindred - Lecture 7
Page 4
Important invariants of a graph
↵(G)
maximum independent set vtcs
↵0(G) maximum matching
edges
pairwise nonadjacent
pairwise disjoint
(G)
minimum vertex cover
vtcs that “cover” all edges
0
minimum edge cover
edges that “cover” all vtcs
(G)
Definition An edge cover of a graph G is a subset of edges
F ✓ E(G) such that for any v 2 V , there exists e 2 F such that
v is an endpoint of e (so v 2 e).
The edge covering number of G is the minimum cardinality of an
edge cover and is denoted 0(G).
Note that
0
(G) is undefined if G has isolated vertices.
Lemma. In a graph G with S ⇢ V (G),
S is an independent set () S is a vertex cover,
and thus, ↵(G) + (G) = |V (G)|.
Proof. ()) No edge has both endpoints in S (since it is an independent set), so every edge of G has at least one endpoint in S. Thus,
S is a vertex cover.
(() Every edge has at least one endpoint in S (since it is a vertex
cover), so there can be no edge between two vertices of S. Hence, S
is an independent set.
⌅
Math 104 - Prof. Kindred - Lecture 7
Page 5
Theorem (Gallai). If G is a graph with no isolated vertices, then
↵0(G) + 0(G) = |V (G)|.
Proof sketch. Let n = |V (G)|.
Given a maximum matching M , construct an edge covering of size
n |M | by adding edges to M to cover unsaturated vertices.
So 0(G)  n ↵0(G).
Given a minimum edge covering F , construct a matching of size
n |F |. Use the idea that for any graph G = (V, E) with minimum
edge covering F , the graph (V, F ) is a disjoint union of |V | |F |
0
nontrivial stars. So ↵0(G) n
(G).
⌅
Corollary. If G has no isolated vertices and is bipartite, then
↵(G) = 0(G).
Proof. We have
↵(G) |{z}
= |V (G)|
previous
lemma
(G) |{z}
= |V (G)|
K-E
thm
↵0(G) |{z}
= 0(G)
Gallai
thm
where we use assumption that G is bipartite to apply the KönigEgerváry theorem and the assumption that G has no isolated vertices
to apply Gallai’s theorem.
⌅
It should be easy to see that ↵(G)  0(G) — each vertex in an
independent set must have its “own” edge to cover it.
Math 104 - Prof. Kindred - Lecture 7
Page 6
Linear programming formulation
⇥
König-Egerváry theorem
1
min vertex
cover problem
Tuesday, February 12, 2013
Math 104, Graph Theory
min vertex
cover problem
1
···
1
n x 1 vector where
i th entry is 1
if i th vertex is in S
⇤
T
min 1 xS
s.t. AT xS
A is n x m
incidence
matrix
⇤
n x 1 vector where
(variable)
i th entry is 1
if i th vertex is in S
A is n x m
incidence
matrix
Linear programming formulation
1
1
min 1T xS
s.t. AT xS 1
xS binary
AT =
⇥
···
1
1
xS binary
vertices
3
7
6
7
6 0 0 1 0 1 07
6
7
4
5
Linear programming formulation
dual of primal linear program
m x 1 vector where
i th entry is 1
if i th edge is in M
max 1T yM
s.t. AyM  1
yM binary
max matching
problem
Theorem (weak duality) Assume that the primal and dual linear programs are both feasible.
Let x be a feasible solution to the primal problem and let y be feasible solution to the dual. Then
objective value of primal ≥ objective value of dual.
dual problem (maximization)
2
edges 6
primal problem (minimization)
Question
What is the relationship between vertex covers and
is theand
relationship
vertex
covers
and
matchings
of primal
a graph?
Theorem (weak duality)
Assume
that What
the
dual linearbetween
programs
are both
feasible.
Question
matchings
ay
graph?
Let x be a feasible solution to the primal problem
andoflet
be feasible solution to the dual. Then
objective value of primal ≥ objective value of dual.
Example
PetersenExample
graph has a vertex cover of size 6 and a matching of size 5
primal problem (minimization)
Petersen graph has a vertex cover of size 6 and a matching of size 5
Proposition. For any graph G, ↵0(G)  (G).
dual problem (maximization)
Proposition. For any graph G, ↵0(G)  (G).
Proof. Let M be a maximum matching of G, and let S be a minimum
vertex covering. For every edge e in M , there exists an endpoint of
Proof. Let M be a maximum matching of G, and let S be a minimum
It follows thateifinwe
findbecause
a feasible
and
feasible
y such that the objective values of
S.can
Thus,
edgesx of
M ashare
no endpoints,
vertex covering. For every edge e in M , there exists an endpoint of
the primal and dual are equal, then
0 those feasible solutions must both be optimal.
↵ (G)
= |M edges
|  |S|of=M (G).
e in S. Thus,
because
share no endpoints,
(certificate
of optimality)
0
↵ (G) = |M |  |S| = (G).
⌅
Corollary. If G = (V, E) is a graph, M ✓ E is a matching of
⌅
G, and S ✓ V is a vertex cover of G such that |S| = |M |, then
Corollary. If G = (V, E) is a graph, M ✓ E is a matching of
• M is maximum,
G, and S ✓ V is a vertex cover of G such that |S| = |M |, then
• S is minimum,
• M is maximum,
• and ↵0(G) = |M | = |S| = (G).
• S is minimum,
Example• and ↵0 (G) = |M | = |S| = (G).
Example
Idea behind proof of König-Egerváry Theorem
Idea behind proof of König-Egerváry Theorem
A
A
B
C
Idea behind proof of König-Egerváry theorem
matching that
saturates A
A
C
B
G[B C]
B
Idea behind proof of König-Egerváry theorem
A
D
G[A
D
D
matching that
saturates A B
D]
C
matching that
saturates B
A
C
D
B
B
A
C
D
B