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Ear Decompositions of Matching Covered Graphs Relat Orio T Ecnico IC{98-03

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Ear Decompositions of Matching Covered Graphs Relat Orio T Ecnico IC{98-03 O conteudo do pre s ente relator io e de unica re sp onsabilidade dos autore s The contents of thi s rep ort are the sole re sp ons ibility of the authors Ear decomp o s itions of matching covere d graphs Marcelo H Carvalho UFMS Campo Grande Brasil Claudio L Lucchesi UNICAMP Campinas Brasil USRMurty University of Waterloo Waterloo Canada Relator io Tecnico IC Fevere iro de Ear decomp o s itions of matching covere d graphs z y Claudio L Lucche s i Marcelo H Carvalho UNICAMP Campinas Bras il UFMS Camp o Grande Bras il x USRMurty Univers ity of Waterlo o Waterlo o Canada February Ab stract Ear decompositions of matching covered graphs are important for understanding their structure By exploiting the properties of the dependence relation introduced by Carvalho and Lucchesi in we are able to provide simple proofs of several wel lknown theorems concerning ear decompositions Our method actual ly provides proofs of generalizations of these theorems For example we show that every matching covered graph G dierent from K has at least edgedisjoint removable ears where is the maximum degree 2 of G This shows that any matching covered graph G has at least ! dierent ear decompositions and thus is a generalization of the fundamental theorem of Lovasz and Plummer establishing the existence of ear decompositions We also show that every C has 2 edges each of which is a removable edge brick G dierent from K and 6 4 in G that is an edge whose deletion from G results in a matching covered graph This generalizes a wel lknown theorem of Lovasz Using this theorem we give a simple proof of another theorem due to Lovasz which says that every nonbipartite matching covered graph has a canonical ear decomposition that is one in which either the third graph in the sequence is an oddsubdivision of K or the fourth graph in the sequence is an odd 4 subdivision of C Our method in fact shows that every nonbipartite matching covered 6 graph has a canonical ear decomposition which is optimal that is one which has as few double ears as possible Most of these results appear in the PhDthesis of the rst author written under the supervision of the second author Intro duction A matching covere d graph i s a nontr ivial connecte d graph in which every e dge i s in some p erfect matching There i s an extens ive literature on matching covere d graphs For a hi story of the sub ject and for all terminology and notation not dene d here we refer the reader to Lovasz and Plummer Lovasz or Murty AMS sub ject class ication C y Supp orte d by UFMS and CAPES Bras il z Supp orte d by grants f rom CNPq and FAPESP x Work partially done dur ing thi s authors vi s it to UNICAMP Bras il f rom Septemb er to Decemb er Carvalho Luccche s i and Murty Two typ e s of decomp os itions of matching covere d graphs namely tight cut decomp os i tions and ear decomp os itions have playe d s ignicant role s in the development of the sub ject For example b oth the s e decomp os itions are e ss ential ingre dients in Lovaszs chracter iza tion of the matching lattice Here we us e tight cut decomp os itions in our study of ear decomp os itions For all the relevant denitions and theorems concer ning the tight cut de comp os ition of a matching covere d graph we refer the reader to the source s cite d ab ove We recall b elow some of the bas ic denitions relate d to ear decomp os itions Let G b e a subgraph of a graph G and let P b e a path in G E G A s ingle e ar of G i s a path P of o dd length in G such that i b oth the ends of P are in V G and ii P i s inter nally di sjoint f rom G The following theorem provide s a decomp os ition of bipartite matching covere d graphs Theorem Given any bipartite matching covered graph G there exists a sequence G G G G 1 2 r of matching covered subgraphs of G where i G K and ii for i r G is 1 2 i+1 the union of G and a single ear of G i i The s equence G G G of subgraphs of G with the ab ove prop ertie s i s an ear de 1 2 r comp os ition of G in which each memb er of the s equence i s a matching covere d subgraph of G Such decomp os itions do not exi st for nonbipartite matching covere d graphs For example K has no ear decomp os ition as in Theorem However every matching covere d graph 4 has an ear decomp os ition with a slight relaxation of the ab ove denition For de scr ibing that decomp os ition we require the notion of a double ear A double e ar of a matching covere d graph G i s a pair of vertex di sjoint s ingle ears of G An e ar of G i s e ither s ingle ear or double ear of G An e ar decomp o s ition of a matching covere d graph G i s a s equence G G G G 1 2 r of matching covere d subgraphs of G where i G K and ii for i r G i s 1 2 i+1 the union of G and an ear s ingle or double of G The following fundamental theorem was i i e stabli she d by Lovasz and Plummer See Theorem The twoe ar Theorem Every matching covered graph has an ear decom position There are var ious pro ofs of thi s theorem The or iginal pro of due to Lovasz and Plummer i s somewhat involve d Later on Little and Rendl gave a s impler pro of Recently another s imple pro of was given by Szigeti A matching covere d subgraph H of a graph G i s nice if every p erfect matching of H extends to a p erfect matching of G It i s easy to s ee that all the subgraphs in an ear decomp os ition of a matching covere d graph must b e nice matching covere d subgraphs of the graph In all the previous approache s an ear decomp os ition of a matching covere d graph Ear decomp os itions of matching covere d graphs G i s e stabli she d by showing how a nice matching covere d prop er subgraph G of G can b e i extende d to a nice matching covere d subgraph G by the addition of a s ingle or double ear i+1 Our approach i s to build an ear decomp os ition of a matching covere d graph in the revers e order To state our theorem preci s ely we nee d to dene the notion of a removable ear in a matching covere d graph Let G b e a matching covere d graph and let P b e a path in G Then P i s said to b e a removable s ingle e ar in G if i P i s an o dd path which i s inter nally di sjoint f rom the re st of the graph and the graph obtaine d f rom G by deleting all the e dge s and inter nal vertice s of P i s matching covere d A removable ear of length one i s a removable e dge The notion of a removable double e ar in G i s s imilarly dene d A removable e ar in G i s e ither a s ingle or double ear which i s removable In trying to e stabli sh the exi stence of ear decomp os itions with sp ecial prop ertie s it i s often convenient to nd the subgraphs in the ear decomp os ition in the revers e order starting with G G Thus after obtaining a subgraph G in the s equence which i s dierent f rom K we r i 2 nd a suitable removable ear s ingle or double and obtain G f rom G by removing that ear i1 i f rom G For example to show that a matching covere d graph G has an ear decomp os ition i it suce s to show that every matching covere d graph dierent f rom K has a removable ear 2 Thi s i s our approach We in f act prove the folowing more general theorem Theorem Any matching covered graph G dierent from K has removable ears single 2 or double Clearly Theorem i s a sp ecial cas e of Theorem In f act shows that every matching covere d graph G has at least dierent ear decomp os itions Supp os e that G G G G 1 2 r i s an ear decomp os ition of a matching covere d graph G If a memb er G of the s equence i s i obtaine d f rom its pre dece ssor G by adding a s ingle double ear then we shall say that i1 G i s obtaine d f rom G by the addition of a s ingle double ear In any ear decomp os ition i i1 of a nonbipartite matching covere d graph G there must b e at least one double ear addition Furthermore if G i s the rst memb er of an ear decomp os ition which i s obtaine d by a double i 1 ear addition then G G are bipartite and G G G are nonbipartite 1 i1 i r In any ear decomp os ition by denition G i s K and G i s an even circuit It i s not 1 2 2 dicult to check that if G i s nonbipartite then it must in f act b e an o dd sub divi s ion of 3 K and if G i s bipartite and G i s nonbipartite then G must b e an o dd sub divi s ion of 4 3 4 4 C We shall refer an ear decomp os ition 6 G G G G 1 2 r of a nonbipartite matching covere d graph G as a canonical e ar decomp o s ition if e ither its third memb er G or its fourth memb er G i s nonbipartite The following fundamental 3 4 theorem was prove d by Lovasz in 1 We only cons ider ear decomp os ition s which are ne By thi s we mean that a double ear P P i s adde d 1 2 to G to obtain G only if ne ither G P nor G P i s matching covere d i i+1 i 1 i 2 Carvalho Luccche s i and Murty Theorem The canonical e ar decomp o s ition Theorem Every nonbipartite match ing covered graph G has a canonical ear decomposition In Lovasz gave s everal applications of the ab ove theorem to extremal problems in graph theory Later on he us e d it to de duce the following theorem which was crucial for hi s work on the matching lattice C Theorem The removable e dge Theorem Every brick dierent from K and 6 4 has a removable edge Lovaszs or iginal pro of of Theorem was quite involve d A s impler pro of was given by Carvalho and Lucche s i Our approach here shall b e to prove Theorem rst and then to der ive f rom it We shall in f act prove the following generalization of Theorem Every brick dierent from K and C has at least removable edges 4 6 A pro of of the ab ove theorem was rst given by Carvalho and Lucche
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