Resultados individuales de los parámetros

In this Section we start describing the new results in this chapter. The class of graphs for which MINMAXLMATCH and MAXINDMATCH are NP-hard to find is extended. The first result deals the hardness of MINMAXLMATCHfor graphs which show some regularity whereas the second one relates MAXINDMATCH on bipartite graphs to another interesting combinatorial problem. Other hardness results will follow as by-products of the results in Section 4.6.

A(Æ;)-graph is a graph with minimum degreeÆand maximum degree. A(d;d)-graph

is a regular graph of degreed(or ad-regular graph). IfP is an NPO graph problem, then(Æ;)-P

(resp.d-P) denotes the same problem when the input is restricted to the being a(Æ;)-graph (resp.

ad-regular graph). Finally, a(Æ;)-graphGis almost regular if=Æis bounded by a constant.

4.3.1

M

IN

M

AXL

M

ATCH

in Almost Regular Bipartite Graphs

The NP-completeness proofs in [HK73] and [YG80] show that it is NP-hard to find a maximal matching of minimum cardinality in a planar cubic graph and in planar bipartite(1;3)-graphs. This

last result can be extended to bipartite(ks;3s)-graphs for every integers >0andk =1;2. The

following result shows how to remove vertices of degree one from a(1;)-graph.

Lemma 29 There is a polynomial time reduction from(1;)-MINMAXLMATCHto(2;)-MIN-

MAXLMATCH.

Proof. Given a(1;3)-graphG, the graphG 0

is obtained by replacing each vertexvof degree one

inGby the gadgetG

vshown in Figure 4.2. The edge

fv;wgincident tovis attached to the vertex v

0. The resulting graph has minimum degree two and maximum degree three. If

M is a maximal

matching inGit is easy to build a maximal matching inG 0 of sizejMj+2jV 1 (G)j jV(M)\V 1 (G)j. For everyv2V 1 (G)addfv 1 ;v 2 gtoM 0

moreover iffu;vg62MthenM 0

will contain also the edge

fv 0

;v 3

g. Conversely every matchingM 0

inG 0

can be transformed into a matchingM 00 =M 00 1 [M 00 2, withjM 00 j jM 0 j, such thatM 00 1

is a maximal matching in GandM 00

is a set of edges entirely contained in the gadgetsG

v. 2 v v 1 2 v 0 v₃ v G

c 1 c 2 a b c d a a b b d d 1 2 1 2 1 2

Figure 4.3: A(1;3)-graph and its 2-padding.

To prove the main hardness result in this section, the following graph operation will be useful.

Definition 20 Thes-padding of a graphG,G

s, is obtained by replacing every vertex

vby a distinct

set of twin verticesv 1 ;:::;v swith fv i ;u j g2E(G s

)if and only iffu;vg2E(G). The vertices of G

s are partitioned into

s layers. Vertices in each layer along with edges

connecting pairs of vertices in the same layer form a copy of the original graphG. For eache = fu;vg 2 E(G)edgese

i = fu

i ;v

i

gfori = 1;:::;sare called twin edges.

Edgese ij =fu i ;v j

gfor eachi6=jare called cross edges. Each copy ofK

s;sobtained by replacing two vertices and an edge in

Gis called a padding

copy ofK

s;sand sometimes denoted by K

e s;s

, to show dependency on the edge in the original graphG.

The following result is a simple consequence of Definition 20.

Lemma 30 IfGis a(Æ;)-graph withnvertices andmedges thenG

sis a

(sÆ;s)-graph with snvertices ands

2

medges.

To prove Theorem 53 it is important to relate(G s

)to(G).

Lemma 31 (G

s

)s(G), for all graphsGands1.

Proof. IfM is a maximal matching inG, a maximal matchingM

sin G

sis obtained by taking the

union ofjMjperfect matchings one in each copy ofK e s;s

withe2M. 2

Lemma 32 (G

s

)s(G), for all bipartite graphsGands1.

Proof. LetG

sbe the padded version of a bipartite graph

G, andM

sbe a maximal matching in G

s.

Thes-weighting of the edges ofGis a functionw:E(G)!f0;:::;sg. For eache2E(G)define w(e)= df jM s \E(K e s;s

1. Ifw(v)= df

fe:v2eg

w(e)thenw(v)2f0;:::;sgfor allv2V(G).

The sum of the weights of the edges incident to v cannot be larger thansotherwise there

would be more thans edges in M

s incident to v

1 ;:::;v

s; therefore one of these vertices

would be incident to more than one edge inM s.

2. Let E(i) = fe 2 E(G) : w(e) = igfor i 2 f0;:::;sg. Then S

s i=1

E(i) is an edge

dominating set ofG(as defined in Section 4.2).

This is true because if there was an edgeeinGnot adjacent to any edge with positive weight

thenK e s;s

would not be covered byM sin

G s.

3. LetG(E(i))be the subgraph ofGinduced byV(E(i)). ThenE(s)is a maximal matching in G(E(s)).

The edges inE(s)must be independent because each of them corresponds to a perfect match-

ing in a padding copy ofK s;s.

4. LetG G(E(i))be the graph obtained by removing fromV(G)all vertices inV(G(E(i)))

and all edges adjacent to them. Then

S s 1 i=1

E(i)is an edge dominating set inG G(E(s)).

5. Letv2V(G G(E(s))); ifw(v)<sthenw(u)=sfor everyu2N(v).

Letv

ibe one of the twin vertices associated with

vthat is not inV(M s

). For eachu2N(v)

each of the edges fv i

;u j

g(forj = 1;:::;s) must be adjacent to a different edge in M s

otherwise they would not be covered.

Using thes-weighting defined above the edges inM

scan be partitioned into

smatchings

each corresponding to a maximal matching inG.

First of all, each edgeeinE(s)corresponds tosdistinct edgese 1 ;:::;e sin M s. Define M(j)=fe j

:for eache2E(s)g. We prove, reasoning by induction ons, that the set of remaining

edges inM s,

M 0

s, can be partitioned into

ssetsM j

such that,M j

[M(j)corresponds to a maximal

matching inG, for eachj=1;:::;s.

BASE. Ifs=2by property 4 above, the setE(1)is formed by a number of paths and even length

cycles,E 1

;:::;E

k. Each cycle of length

2m(for some integerm>1) can be decomposed into two

matchingsM 1

andM 2

of sizemby taking alternating edges. IfE

j is a path then, by property 5

above, neither of its end-points can be adjacent to a vertexvwithw(v) =0. Therefore again two

set of edges are added toM 1

andM 2

STEP. LetH

sbe the graph induced by the edges of positive weight less than s. IfM s is a maximal matching inH s, then w 0

(e)= w(e) 1(resp. w 0 (e) = w(e)) ife 2 M s (resp. e 62 M s ) is an

(s 1)-weighting ofE(G G(E(s)))corresponding to the a maximal matching in thes 1-padding

ofG G(E(s)). The inductive hypothesis applies. 2

Theorem 53 MINMAXLMATCHis NP-hard for almost regular bipartite graphs.

Proof. We will prove hardness for(ks;3s)-graphs, withk= 1ork =2. Yannakakis and Gavril

[YG80] proved that(1;3)-MINMAXLMATCH is NP-hard for bipartite graphs. The hardness of (2;3)-MINMAXLMATCH follows from Lemma 29. Then, k = 1;2the s-padding can be used

to obtain an instance of(ks;3s)-MINMAXLMATCHrestricted to bipartite graphs. The result then

follows from Lemma 31 and Lemma 32. 2

4.3.2

M

AX

I

ND

M

ATCH

and Graph Spanners

Given a graphG=(V;E)a2-spanner is a spanning subgraphG

0

with the property thatdst G

0

(u;v) 2dst

G

(u;v)for everyu;v2V. Lets 2

(G)be the number of edges of a sparsest 2-spanner ofG.

The problem has many applications in areas like distributed computing, computational geometry and biology [PU89, ADJS93]. Peleg and Ullman [PU89] introduced the concept of graph spanners as a means of constructing synchronisers for Hypercubic networks (their results have been recently improved in [DZ]). The problem of finding a 2-spanner with the minimum number of edges is NP- hard [Pm89]. In this Section we present a reduction from the problem of finding a sparsest 2-spanner in a graph to that of finding a largest induced matching in a bipartite graph without small cycles.

For everyGon nvertices and m edges, letB(G) be a bipartite graph with vertex sets U =fu

e

:e2 E(G)gandW =fw C

:Cis a cycle of length 3 inGg. Two verticesu e

2U and w

C

2WinB(G)are adjacent if the edgeebelongs to the cycleCinG.

Lemma 33 s

2

(G)m I

(B(G)).

Proof. LetM be an induced matching inB(G). DefineS =fe2E(G) :fu

e ;w

C

g62Mg. We

claim thatSis a spanner inG. This is so because for everyfu e

;w C

g2Mthe edgesf;g 2E(G)

that form the cycleCalong witheare such thatfu f ;w C g;fu g ;w C

gcannot be inMand therefore

aref;g2S. 2 Lemma 34 s 2 (G)m I (B(G)).

Proof. LetG 0

be a 2-spanner inG. We prove that we can construct an induced matching inB(G).

Ife2E(G)nE(G 0

)then there exist two edgesf;g 2G 0

such thatC =fe;f;ggis a triangle in G. We addfu

e ;w

C

gto the matching inB(G)and we say that the triangleCcoverse. LetM be

the set of edges inE(B(G))constructed in this way. Since a triangle can only cover one edge, there

are no two edges inMsharing a vertexw

C. Also by our construction every edge in

E(G)nE(G 0

)

is considered only once so that there are no two edges inM sharing an edge-vertex. We claim

thatMis an induced matching inB(G). Letfu e

;w C

g2 M, assume edgeebelongs to triangles C

1 ;:::;C

t(e), and let

C=fe;f;gg. No edgefu h

;w C

j

gcan be inMbecause, by the definition of M,fu

h ;w

Cj

g2M would imply thate2E(G 0 )and thereforefu e ;w C g62M. Similarly neither u f, nor u gcan be in V(M). 2

The girth of a graph is the length of its shortest cycles.

Theorem 54 MAXINDMATCHis NP-hard on bipartite graphs with girth at least six.

Proof. For every graphG, the girth ofB(G)is at least six, since no two verticesu e

;u f

2 U can

share two neighbours inW. The result follows from Lemma 33 and 34. 2

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