A tree version of Konig's theorem

6 Pages·0.12 MB·English

by  Ron Aharoni

A TREE VERSION OF KO¨NIG’S THEOREM
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0 RON AHARONI, ELI BERGER, AND RAN ZIV
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2

n
a
J Abstract. Ko¨nig’stheoremstates thatthecoveringnumberandthematch-
ingnumberofabipartitegraphareequal. Weproveageneralisation,inwhich
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the point inone fixed sideof the graph of each edge is replaced by a subtree
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ofagiventree. TheproofusesarecentextensionofHall’stheoremtofamilies

ofhypergraphs,bythefirstauthorandP.Haxell[3].
]
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C
1. Width and matching width of families of trees
.
h
A matching in a hypergraph is a set of disjoint edges. In [3] (see also [2] and
t
a [5]) there were introduceda few notions which provedto be fruitful in the study of
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matchings. The two main ones are those of width and matching width, which are
[ presented below.

LetH andF betwohypergraphsonthesamevertexset. AsubsetC ofF issaid
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v to be H-covering (or “anH-covering”) if every edge in H meets some edge from C
4 (in other words, the union of C is a vertex cover for H). The F-width w(H,F) of
3 H isthe minimalsizeofanH-coveringsetofedgesfromF. The F-matching width
1
mw(H,F) of H is the maximum, over all matchings M in H, of w(M,H). We
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write w(H) and mw(H) for w(H,H) and mw(H,H), respectively, and call these
1
9 parameters the width and matching width of H. These notions have counterparts
9 in which the covering set is required to be a matching. The independent F-width
/ iw(H,F) of H is the minimal size of a covering matching in F. The independent
h
t matching width imw(H,F) of H is the maximum, over all matchings M in H, of
a
iw(M,F). Again, we write iw(H) for iw(H,H) and imw(H) for imw(H,H), and
m
callthemtheindependent widthandindependent matchingwidthofH,respectively.
:
v (The notion of independent matching width is used in [3], but is not given there a
i special name.)
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In [3] the following theorem was proved:
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a
Theorem 1.1. If A is a family of hypergraphs such that mw(SB)≥|B| for every
B ⊆A, then there is a choice of disjoint edges, one from each hypergraph in A.
Clearly, mw ≤ w and imw ≤ iw, and in general strict inequalities occur. How-
ever, as we shall prove below, for families of trees equality holds in both. In fact,
both equalities are special cases of a more general result. For two hypergraphs
H ,H andasymmetricrelation∼onH wedenotebyw(H ,H , ∼)the minimal
1 2 2 1 2
size of a ∼-related subset of H which is an H -covering. By mw(H ,H , ∼) we
2 1 1 2
denote the maximum, over all matchings M in H , of w(M,H , ∼).
1 2
The research of the first author was supported by the fund for the promotion of research at
theTechnion.
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2 RON AHARONI, ELI BERGER, AND RAN ZIV
Theorem 1.2. Let H ⊆ H be families of subtrees of a given tree. Let also ∼
1 2
be a symmetric relation on H containing the disjointness relation (i.e, if x,y are
2
disjoint members of H then x∼y). Then
2
mw(H ,H , ∼)=w(H ,H , ∼)
1 2 1 2
The equality mw(H) = w(H) for a family H of trees is obtained upon taking
H =H =H andtaking∼tobethetotalrelation(i.e.,x∼yforallx,y ∈H). The
1 2
equality imw = iw is obtained upon taking H = H = H and ∼ the disjointness
1 2
relation.
The theorem will clearly follow from the following lemma:
Lemma 1.3. LetH ,H and∼beasabove, andletc,dbetwointersectingedges of
1 2
H . Then one of w(H \{c},H ,∼)or w(H \{d},H ,∼) is equalto w(H ,H ,∼).
1 1 2 1 2 1 2
(Once the lemma is proved, the theorem follows by deleting edges from H one
1
by one, until a matching is obtained).
Proof of the Lemma Write w = w(H ,H ,∼). Suppose, for contradiction, that
1 2
w(H \{c},H ,∼) = w(H \{d},H ,∼) = w−1. Let C be a ∼-related covering
1 2 1 2
of size w −1 of H \{d} and D a ∼-related covering of size w −1 of H \{c}.
1 1
Clearly, D does not meet c and C does not meet d. Let S be the connected part
1
of T \(V(c)∩S{V(t):t∈C}) containing d and let S =T \V(S ). (Here T \A,
2 1
whereAis a subsetofV(T),denotes the forestobtainedby deleting the verticesof
A from T.)
Let C = {g ∈C : g ⊆ S } (i = 1,2) and D = {g ∈ D :g ⊆ S } (i = 1,2). Let
i i i 1
also K =C ∪D ∪{c} and L=C ∪D .
1 2 2 1
Obviously, K is an H -covering, since the only members of H not covered by
1 1
C andD arethosewhichmeetbothS andS ,andthese arecoveredbyc. Since
1 2 1 2
D does not meet c, the tree c is ∼-related to all members of D . Also, by the
2
definition of C , no tree from C meets c, and hence c∼x for every x∈C . Thus
1 1 1
K is ∼-related. A similar argument shows that also L is a ∼-related H -covering.
1
But |K|+|L| = 2w−1, and hence either |K| < w or |L| < w, which contradicts
the definition of w.
2. A Ko¨nig-like theorem
AhypergraphH iscalledapoint-treehypergraphifitisobtainedfromabipartite
graph by replacing, in each edge, the point in one side of the graph (the same side
foralledges)byatree. Moreformally,H ispoint-treeif,forsomesetX andatree
T whose vertex set is disjoint from X, each edge e ∈ H is of the form {x}∪V(t),
wherex=x(e)∈X andt=t(e)isasubtreeofT. Forsuchahypergraphwedenote
by σ(H) the number w(H,F), where F = F(H) = {{x(e)} : e ∈ H}∪{V(t(e)) :
e∈H}. (That is, the edges of F are to be caught by vertices in X, or trees t(e)).
Recall that the matching numberν(H) of a hypergraphH is the maximalsize of a
matching in H. We shall prove the following generalisationof K¨onig’s theorem:
Theorem 2.1. σ(H)≤ν(H) for every point-tree hypergraph H.
K¨onig’s theorem is the special case in which all trees t(e) are singletons.
In [6] (see also [4]) there were considered objects which are similar to point-tree
hypergraphs: bipartite graphs in which each point (on both sides of the graph) in
each edge is replaced by an interval. In [6] it was proved that τ ≤ 2ν for such
A TREE VERSION OF KO¨NIG’S THEOREM 3
hypergraphs. It is interesting to note that the proof there was topological (as was
thesimplerproofin[4]),andatbasesoistheproofhere,sinceitreliesonTheorem
1.1, whose proof is topological.
Infact,weshallneedastrongerversionofTheorem1.1,its“deficiencyversion”.
Definition 2.2. Thedeficiencydef(A)ofafamilyAofhypergraphsistheminimal
natural number d such that mw(SB)≥|B|−d for every subfamily B of A.
The following strengthening of Theorem 1.1 follows from it by a standard argu-
ment (see [1]).
Theorem 2.3. Every family A of hypergraphs has a subfamily D of size at most
def(A), such that A\D has a choice function of disjoint edges.
Proof of Theorem 2.1 For each x ∈ X let K(x) be the hypergraph {V(t(e)) :
x(e) = x}. Applying Theorem 2.3 to the family {K(x) : x ∈ X} yields the ex-
istence of a subset Y of X such that, writing K = S{K(y) : y ∈ Y}, we have
mw(K) = |Y|−(|X|−ν(H)). By Lemma 3.1 w(K) = mw(K). Let C be a K-
coveringsetofedgesinK ofsize|Y|−(|X|−ν(H)). ThenC∪X\Y isanH-covering
set consisting of elements of F(H), of size ν(H). This proves the theorem. 2
3. The special case of families of intervals
Aspecialcaseoftreesisthatofintervals. Thus,alltheresultsaboveapplytothe
specialcase of families of intervals. But it turns out that in this case the equalities
mw = w and imw = iw can be strengthened. For example, the matchings which
are “hard to catch” can be described explicitly, and can be found efficiently. To
show this, we shall need the following notation.
WesaythatasetofverticesinagraphGisk-independentifthedistancebetween
any two of its members is larger than k. (Thus 1-independence is just the usual
independence.) By γ (G) we denote the maximal size of a k-independent set in G.
k
AsetofedgesinahypergraphH isk-remoteifitisk-independentinthelinegraph
L(H). We write ζ (H) for γ (L(H)). Clearly, for all hypergraphs we have
k k
(1) ζ ≤mw≤w
2
Here we shall prove:
Lemma 3.1. For a family F of intervals on the real line equality holds throughout
(1), namely ζ (F)=mw(F)=w(F).
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Given a family of intervals F, we denote by ℓ(F) the leftmost interval in F,
namely the one with minimal right endpoint. For a real number x we denote by
F(>x) the set of those intervals in F which lie entirely within (x,∞).
Proof ofLemma3.1By(1)thelemmawillbeprovedifwecanexhibita2-remote
setRandacoveringsetC ofthesamesize. Weconstructsuchsetsbyaninductive
process.
Letr =ℓ(F),andc =[x ,y ]anelementofF meetingr whoserightendpoint
1 1 1 1 1
is largest. Let now r = ℓ(F(> y )), and c = [x ,y ] the interval in F meeting
2 1 2 2 2
r whose right endpoint is maximal. (Note that c may intersect c .) In general,
2 2 1
having defined c = [x ,y ] (i > 0), we let r = ℓ(F(> y )) and choose c as
i i i i+1 i i+1
4 RON AHARONI, ELI BERGER, AND RAN ZIV
the interval in F meeting r and extending furthest to the right. The process
i+1
terminates when, at some stage t, there is no interval lying entirely to the right of
c . Then, clearly, R={r ,... ,r } is remote, while C ={c ,... ,c } is covering. 2
t 1 t 1 t
A similar equality can be proved for ζ for all k. For any graph G we denote
k
by ρ (G) the minimal number of vertex sets of diameter at most k, whose union
k
is V(G). Obviously, ρ (G) ≥ γ (G) for all k. One corollary of Lemma 3.1 is that
k k
in an interval graph ρ (G) = γ (G) (in fact, it implies something even stronger,
2 2
namely that an interval graph G can be partitioned into γ (G) sets of radius 1.)
2
This equality is true for all k, that is, in an interval graph ρ (G) = γ (G) for all
k k
k. We shall prove this for a more general class of graphs, that of incomparability
graphs. A graphis called anincomparability graph if there exists a partialorder on
its vertex set, such that two vertices in the graph are connected if and only if they
areincomparable. As is wellknown,anintervalgraphisanincomparabilitygraph,
the order on the intervals being that of “lying completely to the right of”.
Proposition 3.2. In an incomparability graph ρ (G)=γ (G) for all k.
k k
Given a graph G and a natural number k, we denote by G∗k the graph with
the same vertex set as G, in which two vertices are connected if and only if their
distance inG does notexceedk. Obviously,γ (G)=γ(G∗k), andρ (G)=ρ(G∗k).
k k
Thus, if G∗k is perfect, then G satisfies Proposition 3.2. Since incomparability
graphs are known to be perfect, Proposition 3.2 will follow from:
Lemma 3.3. If G is an incomparability graph then so is G∗k.
Proof Let “≻” be a partial order on V(G) such that x and y are connected in
G if and only if they are ≻-incomparable. Let “=” be the relation defined by x=y
if and only if x≻y and (x,y) 6∈ E(G∗k). Clearly, it suffices to show that “=” is a
partial order.
Assumethatitisnot. Thenthereexistx,y,z ∈V(G)suchthatx=y, y=z,while
x 6 =z. There exist then vertices x = u ,u ,u ,... ,u = z (t ≤ k) such that u is
0 1 2 t i
≻-incomparable to u for all 0≤i<t. Since x=y, all u must be ≻-comparable
i+1 i
to y (or else the sequence u ,... ,u ,y would show that (x,y) ∈ E(G∗k)). Since
0 i
u ≻y and y≻u , it follows that there exists i<t such that u ≻y and y≻u . But
0 t i i+1
then, by the transitivity of the relation ≻, we have u ≻u , a contradiction. 2
i i+1
Our next observation on the interval case is that the equality imw=iw can be
given in this case a constructive (and algorithmically efficient) proof.
Theorem 3.4. imw(F) = iw(F) for every hypergraph F of intervals. Moreover,
thereisapolynomial timealgorithm producingamatchingM suchthatiw(M,F)=
iw(F).
Proof Writek=iw(F). Weshallprovethetheorembyconstructingamatching
M in F satisfying:
(P) Every matching of size k−1 in F misses an interval from M.
Note, first, thatwe canassume thatno two endpoints ofintervals inF coincide:
if there is such coincidence, we can enlarge some intervals, without changing the
intersection pattern of the family. Henceforth we shall make this assumption.
ForamatchingE inF wewriteE =(e ,... ,e )ife arethe edgesofE ordered
1 t i
from left to right. Let E be such a matching, and write e = [a ,b ], j ≤ t. For
j j j
A TREE VERSION OF KO¨NIG’S THEOREM 5
each j ≤ t we denote by dj(E) the interval ℓ(F(> bj−1)) (where b0 is defined as
−∞, meaning that d (E) is always ℓ(F)). If F(> b ) is non-empty, we write also
1 t
d (E)=ℓ(F(>b )). Note that d (E) meets e if and only if there is no interval
t+1 t j j
from F lying between ej−1 and ej; or, in the case j = 1, if and only if there is no
interval in F lying to the left of e . By m(E) we denote the maximal index m≤t
1
such that there is no interval from F lying between ej−1 and ej for all j ≤ m (if
there is an interval from F to the left of e , we write m(E) = 1). If m(E) = t,
1
we say that E is dense. Note that if E misses some edge from F then it misses
d (E).
m(E)
Let L be the set of all right endpoints of intervals e in some dense matching
i i
E = (e ,... ,e ). An interval [a,b] is called L -free if [a,b]∩L ⊆ {b}. Let M be
1 t i i
the set of all intervals in F which are equal to d (E) for some dense matching E,
i
and are L -free for all j ≤i.
j
The proof will be complete if we show that M is a matching, and that it satisfies
property (P).
Let us first show that M is a matching. Suppose, for contradiction, that d =
d (E)=[a,b] and d′ =d (E′)=[a′,b′] are two intersecting members of M, where
p q
E =(e1,... ,ep−1) and E′ = (e′1,... ,e′q−1) are two dense matchings. By assump-
tion b 6= b′, so we may assume (say) that b < b′. Note that b ∈ L , since the
p
matching (e1,...ep−1,d) is dense. Since d′ ∈M, this implies that q <p. Let y be
the right endpoint of e′q−1. Since d ∈M and y ∈Lp−1, it follows that y < a. But
then the interval d = [a,b] witnesses the fact that d′ 6= ℓ(F(> y)), contradicting
the definition of d′.
NextweshowthatM satisfiesproperty(P).Supposethatthereexistsamatching
Z = (z : j ≤ t) in F meeting all intervals in M, where t < k. Among all such
j
matchings Z, choose one in which the right endpoint of z is maximal. Write
m(Z)
m = m(Z) for this Z. Since iw(F) = k, the matching Z misses an interval from
F. This means that Z misses d = d (Z). If d ∈ M, we are done. If not, then d
m
contains a point from L for some j ≤ m. That is, it is the right endpoint of an
j
interval w in some dense matching W = (w ,... ,w ). But then the matching
j 1 m
(w ,... ,w ,z ,... ,z ),likeZ,doesnotmissanyedgefromM. Thiscontradicts
1 j m+1 t
the maximality property of Z.
To see that M above can be constructed in polynomial time, it suffices to note
that the sets D of those intervals which appear as the j-th element in some dense
j
matching can be constructed inductively in plynomial time, and that M is defined
by these sets. 2
References
[1] R.Aharoni,Ryser’sconjecturefortripartitehypergraphs,Combinatorica, toappear
[2] R.AharoniandM.Chudnovsky, Specialtriangulationsofthesimplexandsystemsofdisjoint
representatives, submitted for publication
[3] R.AharoniandP.Haxell,Hall’stheoremforhypergraphs, Jour. Graph Th.,toappear
[4] T.Kaiser,Transversalsofd-intervals,Discrete.Comput. Geom. 18 (1997), 195-203.
[5] R.Meshulam,Thecliquecomplexandhypergraphmatching,Combinatorica, toappear
[6] G.Tardos,Transversalsof2-intervals,atopologicalapproach,Combinatorica15(1995), 123-
134.
6 RON AHARONI, ELI BERGER, AND RAN ZIV
Departmentof Mathematics,Technion, Haifa,Israel 32000
E-mail address, RonAharoni: [email protected]
E-mail address, EliBerger: [email protected]
Departmentof ComputerScience, Tel-Haicollege, UpperGalilee,Israel12210
E-mail address, RanZiv: [email protected]