A Graph Invariant and 2-factorizations of a graph

A Graph Invariant and 2 -factorizations of a graph Xie Y ingtai Chengdu University (xtetai1@sina.com.cn) Abstract A spanning sub graph of a graph G is called a [0 ,2]-factor of G, if 0 ( ) 2 dx  for () x V G  . is a union of so me disjoint c y cles , paths and isolate vertices , that span the graph G. It is easy to get a [0,2] -factor of G and there would be many o f [0,2] -factors for a G. A characteristic number for a [0 ,2]-factor, which reflect the number of the paths and isolate vertices in it, . T he [0,2]-factor of G is ca lled maximum i f i ts c haracteristic number i s minimum, a nd is called characteristic nu mber of G.It to be p roved that charac teristic number of graph is a graph invariant and a polynomial ti me algorithm for co m puting a maxi mum [0,2] -factor of a graph G has been given in this paper. A [0,2]-factor is Called a 2-factor , if its characteristic number is zero. T hat is ,a 2 -factor is a set of some disjoint cy cles, that span G. A polyno mial ti m e algor ism for com p uting 2-facto r from a [0,2]-factor, w hic h can be got ea sily, is given.. A HAMILTON Cycle is a 2-factor, therefore a necessary condition of a HAMILT ON Graph is that, the grap h contains a 2-factor or the character istic number o f the graph is zero . The algoris m , given in this paper, makes it po ssible to examine the condition i n polynomial time. Key words: 2- factor , [0,2]-factor ,Alternate Chain, P-chain , characteristi c number of graph, graph invar iant 1. Introduction A 2-factor of Graph G is a set of disjoint cycles that span G . 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [1][2][3][4] A Hamilt onian cycle is then a 2-fact or, and in one sense, it is the simpl est 2-factor as it is composed of a single cycle. In another sense, it may be the most difficult 2-factor to find, as we must force a single cy cle. . To our knowledge ,there are no effici ent A lg orithm for finding the 2-fact or in general graph. In some cases an algorithm that com putes such a 2-factor is also given. One such algori thm is due to Petersen. Petersen' s result establishes the existence of 2-factors in 2 m -regul ar graphs only. Gopi and Epstein [5] propose an algorithm to compute 2-factors of 3-regular graphs. Their algorithm computes a perfect matching of the input graph. The edges, that is not in the computed matching define a 2 - factor. Diaz- G utier rez and Gopi [4] present two di ff erent methods to compute a 2-f actors of graphs of maxim u m degree 4. The method c onsists of first computing a perfect matching on the input graph, after rem oving the edges in the m atching from the input graph, and comput ing a new matchi ng on the remaining subgraph. The 2- fa ctor is defined by the edges in the union of both perfect matching. All appearance, their algorithm does not work on graphs with an odd number of verti ces even when these are 4-regular. In fact, there are graphs with an even number of vertices where this algorithm also fails. The secon d method by Diaz -Guti errez and Gopi is called the templ ate substitution algorithm. In this method, vertices of degree 4 are replaced by templates, constructing by six vertex, to obtain an inflated graph. A perfect mat ching of the in fl ated graph can be translated into a 2-factor of the input graph given that exactly two outside vertices of a templ ate connect to vertices of other templates. This algorithm is efficient for 4-regular only although the authors claim that their templates can also repl ace v ertices of degr ee less than 4; however, no det ails are provided. Um ans [6] presented a known algor ithm for computing 2-factors of a general graph. The algorithm is based on linear programming and, alt hough sim ple to describe (as a set of linear equations), it is a ff ected by the underly ing complexity of linear programmi ng ,but unfortunately, the problem about complexity of l inear programming is yet a open problem [7][3] . We will propose a poly n om ial time algorithm for computing 2-factor of general graph. F ro m any of a easily accessible [0,2]- factor to compute out the 2-f actor in stages. This algori thm using a namely P-chain with respect to a [0,2]- factor of G is similar to augm enting path algogorism in the matchi ng problem. 2. Basic theorem The graph considered in this paper will be fini te, undirect ed and simple. Let G ( , ) VE  is a graph on n vert ices with v ertex set () VG ( | ( ) | V G n  ) and edge set () EG . Let S is a subgraph of G , () x V G  ,the () S dx is the degree of vertex x in S. 12 ( , , ... ) k P x x x is a path with two end verti ces 1 , k xx and inner vertices ( 1 ) i x i k  , when ij  , ij xx  . 1 ( ) 1 , ( ) 1 P p k d x d x  and ( ) 2( 1 ) Pi d x i k    .. 1 2 1 ( , , .... , ) k C x x x x is a cycle with vertices 12 , , .. . k x x x , ( ) 2( 1 , 2 , ..., ) Ci d x i k  .when kn  , 12 ( , , ... ) n P x x x is called HAMIL TON path (H path), 1 2 1 ( , , ... , ) n C x x x x is called HAMIL TON cy cle (H cycl e). Let ( , ) e x y  is a edge with end vertices , xy .Two edges 12 , ee is called adjacent if they have one (only one) common vert ex. A sequence of adjacent edges without repeat 12 ... k L e e e  is called a chain ( a close chai n if 1 , k ee is adjacent to each other, a open chain in the othe rwise). Let 1 ( , )( 1 , 2 , .. . 1 ) i i i e x x i k     then 1 2 1 ( , , ... , , .. . ) i i k L x x x x x  is a chain and 1 i x  is the common vertex of edges 1 , ii ee  . A path is also a chain, but a chain would not be necessarily a p ath,for example, i n F ig.1(a), the ( 1 , 5 , 4, 8 , 9 , 6 , 5 ,12) L is a chain only , but is not a path. Definition 2.1 Let is a spa nning sub graph of graph G and 0 ( ) 2 dx  for () x V G  , we call a [0,2]- factor of G. is a union of disjoint cycles , paths and isolated verti ces that span G. L et () ( ) ( 2 ( )) s v V G T d v     (2.1) Then ( ) S T is a even and ( ) / 2 S T is the number of pat hs and the isol ated v ertices in .The ( ) S T is called characteristic number of . When ( ) S T =0 is 2-factor of G and If is a single cycle then is a H cycle. If G is a connected graph and ( ) 2 S T  then there m ust be a H-pat h in G. Example 1 In Fig.1,the two paths 1 ( 1 , 2 , 3 , 4, 5 , 6 , 7, 8 , 9 ,10 ,11 ) P , 2 ( 12 ,13 ,14 , 1 5 ) P and a vertex { 16 } in G form a [0,2]-fact or, that 1 2 { 1 6 } P P   is a [0,2]- factor of G. And ( ) 6 S T  . We here give a brief explana tion for following work. F irst, I t is easy to get a [0,2]-factor of G, for example ,the null graph of G(graph with V(G),but without any edges)is a namely [0,2]- factor. Let 1 is a [0,2]-factor of G, we will propose an Algorithm to get a [0,2]-factor 2 from 1 , and 21 ( ) ( ) 2 S S T T  ,...,getting 1 i  from i , and 1 ( ) ( ) 2 S i S i T T   , until k with ( ) 0 s k T  is a 2- factor of G. For this purpose, we need the following conception: we consider a graph G with a given partiti on of E(G) into disjoint subs E 1 and E 2 , a chain 12 ... k L e e e  is called alternating, if the adjacent edges in L are alternately in E 1 and E 2 . Definition 2.2 L et is a [0,2]- factor of G. A chain 1 2 1 ... ... i i k L e e e e e   is cal led P-chain with respect to , if L is a alter nating respect to com plementary subs E( ) and E(G) \ E(R),and satisfyi ng following condi tion: (1) 1 , ( ) \ ( ) k e e E G E  . (2 ) when L is open then there are a end vertex u of 1 e and a end vertex v of k e such that ( ) 1 , ( ) 1 d u d v  . (3) when L is close then 1 e and k e with common end v ertex u and ( ) 0 du  . Special ly, we al so consider the degenerate chai n, when 1 k  . Intui tively speaking, a P-chain is a alt ernating chain, connecting two verti ces , uv which degree in is less than two. and the two end edges do not belong to ( ) E ,that is ,to ask 1 , ( ) \ ( ) k e e E G E  . The P-chain play s very important role in our algorithm to be established ,so we explain it more i n details. Let 1 ( , )( 1 , 2,... ) i i i e x x i k   and 1 , ii ee  with comm on vertex 1 i x  then 1 2 1 ( , , .... ) k L x x x  is a P-chain w ith k edges i e ( 1 , 2 , ... ik  ), if it i s satisfyi ng: (1) k m ust be a odd. when j is odd ( ) \ ( ) j e E G E  ,when j is even () j e E R  . (2) if 11 k xx   then 11 ( ) 1 , ( ) 1 k d x d x   else if 11 k x x u   then ( ) 0 du  . ( ) 2( 1 1 ) j d x j k     . Because ther e are only two edges in with comm on vertex j x for every ( 1 1 ) j x j k    , so a P-chain L pass j x twice at most , t herefor, (3) ( ) 2, 4( 1 1 ) Lj d x j k     . Example 2 : In example 1(Fig.1), we have [0,2]- factor 1 2 { 1 6 } P P   ,then the alternate chains, 12 ( 1 , 5 , 4, 8 , 9 , 6 , 5 ,12) , ( 16 , 2 , 3 , 1 4,1 3 ,16 ) LL are the P -chai n with respect to , 1 L is open and 2 L is close. Special ly, the edge (11,15) i s also a P- chain with r espect to ,t hat is degenera te. We will t o prove that if there is a [0,2]-factor of G and a P-chain w ith respect to then we can find a [0,2] -factor  from and the num ber of paths and isolated vertices in  is less than the num ber of which in . This is what t he Theorem 2.1 below cl aims. Theorem 2.1 Let is a [0,2]-factor of G, T S ( )>0, and 1 2 1 ( , , ... , ) kk L x x x x  ( 1 ( , ) i i i e x x   ) is a P-chain w ith respect to then L    1 Is also a [0,2]-factor of G and T S (  )=T S ( ) - 2 Proof : Fi rs t we prov e that 0 ( ) 2 dx   for () x V G  . If () x V L  then ( ) ( ) d x d x   ,therefor, 0 ( ) 2 dx   . If () x V L  then 1 G 1  G 2 is a graph (G 1  G 2 )=V(G 1 )  V(G 2 ), E(G 1  G 2 )=E(G 1 )  E(G 2 ) (A  B=(A  B) \ (A  B)) Case 1): We first consider the two end vertices 11 , k xx  of L, it can be easily seen from definiti on 2.2 that 1 1 ( ) ( ) 1 d x d x   and 11 ( ) ( ) 1 k k d x d x     when 11 k xx   (For example the vertices 1 and 12 in Fig .1) and 11 ( ) ( ) 2 k d x d x     when 11 k xx   (For exam pl e the vertex 16 in F ig. 1) . Case 2): ( 1 , 1 ) i x x i k    then ( ) 2, 4 Li dx  2 a ) I f ( ) 2 Li dx  and ( ) 2 i dx  then there are three edges with common vertex i x . One and only one among the three edges belongs t o ( ) ( ) E L E  ,threfor ( ) 2 i dx   (for exam ple the vert ices 4,8,9,6 in F ig.1) . 2 b ) If ( ) 4 Li dx  and ( ) 2 i dx  then there are four edges with common vert ex i x .Two and only two among the four edge belongs to ( ) ( ) E L E  ,theref or ( ) 2 i dx   (for example t he vertices 5 in Fig.1) . we have proved that ( ) ( ) d x d x   for () x V G  except 1 x and 1 k x  .And or 11 ( ) ( ) 2 k d x d x     (when 11 k xx   ,in this cas e 11 ( ) ( ) 0 k d x d x   ) or 1 1 ( ) ( ) 1 d x d x   and 11 ( ) ( ) 1 k k d x d x     (when 11 k xx   ), as a result of case 1 in proof .Hence one can to conclude that:  is al so a [0,2]- factor of G and T S (  )=T S ( ) - 2 . Theorem has been proved. □ Example 3. I n fig.1,by example 1,exam ple 2,let 11 L  ,then 1 ( 1 5 ,14 , 1 3 , 1 2 , 5 ,1 , 2 , 3 , 4, 8 , 7 , 6 , 9,1 0 ,11 ) { 1 6 } P   is also a [0,2]-factor of G. A nd 1 ( ) ( ) 2 4 S S T T    . And the edge (11,15) is a P -chai n with respect to 1 , therefor the [ 0,2]-fact or of G 21 ( 1 1 , 15 ) ( 15 ,14 , 1 3 ,12 , 5 ,1 , 2 , 3 , 8 , 7, 6 , 9,1 0 ,11 ,1 5 ) { 1 6 } C     Formed by a cy cle and a isolat ed vertex and 2 ( ) 2 S T  . And the chain 2 ( 16 , 2, 3 ,14 ,13 ,1 6) L is also a P -chai n with respect to 2 , then 3 2 2 1 2 ( 1 , 5 ,1 2 ,13 ,1 6 , 2 ,1 ) (3 , 1 4 ,15 ,1 1 ,1 0 , 9 , 6 , 7, 8 , 4 , 3 ) L C C     Is a 2-f actor of G,as shown i n fig.1 (b). Definition 2.3 The [0,2]-factor of G is maximum , if which characteristi c number ( ) S T is minimum. The characterist ic number of a maxim um [0.2] -factor is called charact eristic number of Graph G. There can be no P - chain with respect a m aximum [0,2]-factor since such a P - chain can be used, by Theorem 2.1,to get a [0,2]-factor with small er Characteristi c number. It turns out that the converse is true as w ell. Theorem 2.2 A [0,2]-factor of G is maxim um if and only if there is no P -chain w ith respect to . Proof : One direction follow s from Theorem 2.1.For other direction ,we suppose that there is no P-chai n in G with respect t o , and yet is not m aximum. That i s, there is a [0,2]-factor  of G such that ( ) ( ) SS TT  ,we consider the edges in  ; these edges toget her with their end vertices form a subgraph ( , ) V    of G (this subgraph may be disconnected) .We will to prove t hat there m ust be a P-chain with respect to in (thus,in G) . The main idea of proving theorem 2.2 is coming from the proof of similar theorem in theory of maximum matchi ngs, but here is more com plicated and m ore difficult. First ,we have ( ) ( ) ( ) ( ) ( ) ( ) | ( ) | | ( ) | S S v V G v V G T T d v d v E E           Thus | ( ) | | ( ) ( ) | | ( ) | | ( ) ( ) | E E E E E E         (2.2) Let ( ) ( ) \ ( ) ( ) E E E E    ( ) ( ) \ ( ) ( ) E E E E      then E( ) is parted into two disj oint subs ( ) E and () E  . Abov e inequati on shows that | ( ) | | ( ) | E E   . This is a k ey f act to be used for our proof. Those verti ces u in ( ) V are called terminus if ( ) 1 o r ( ) 1 d u d u   . Let u is a terminus and ( ) 1 du   ,then the edge ( , ) ( ) e u v E  with u as a end vertex is called - end- edge . Similar , a  -end- edge e is such a edge that () eE  with u as a end v ertex and ( ) 1 du  . A end-edg e either is a - end- edge or is a  - end- edge . Those vertices u in ( ) V are called inner v ertices if its degree is 2 in both and  (i.e. ( ) 2 du  and ( ) 2 du   ). Obviously , if u is a inner vertex then or ( ) 2 du  and there is a twain alternate edge w ith common vertex u or ( ) 4 du  and there are two twain alter nate edges with com mon vertex u . Another kin d of vertices u are such that ( ) 3 du  ,one can descry that in this case u is a term inus for a end-edge and is also a inner vertex f or a twain alternate edges . Let 12 ... k L e e e  is a alternate chain with respect to ( ) E and () E  in with two end -edges 1 e and k e (We also c onsider the deg enerate, that is k= 1) . Now to prov e that can be decomposed ♀ into abov e alternate chains of . In fact, let 1 x is a ter m inus (It must exist in ( ) V because ( ) ( ) SS TT  )then there is a end- edge 1 1 2 ( , ) e x x  ,if 2 x is also a term inus then 1 e is already a alternate chain(degenerate),if 2 x is a inner vertex then there is a alternate (with 1 e ) edge 2 2 3 ( , ) e x x  ,preceding in this fashion, we can get a al ternate chain 1 1 2 ... k L e e e  with two end- edges 1 , k ee .we consider the e dges in 1 L  ; these edges togeth er with their end vert ices form a subgraph 1 1 1 ( , ) VL  of G (this subgraph may be disconnected). I t can be easily validated that every terminus in 1 is also a terminus in and every inner vertex in 1 is also a inn er vertex in .Thus we can ditto get a alter nate chain 12 2 2 2 2 ... k L e e e  with two end- edge 1 22 , k ee in 1 (but is also in ). Proceeding in this fashi on ,we can get: 0 1 2 , , , . . . m   and 12 , , ... m L L L such that 1 ( , ) i i i i VL   And 1 i L  is a alternating chain in i (Thus, is also in ).We have already decom posed into m alternate chains with two end-edge, by this wa y . A alternate chain L with two end-edges have only one among following t hree form: (1) Two en d-edge of L are  - end - edge. (2) Two en d-edge of L are - end - edge. (3) One of two end-edges of L is  - end - edge and ano ther is - end - edge. Let L  to denot e the length of L , L denote the num ber of edges belonging to ( ) E and L  denote the number of edges belonging to () E  . In form (1) the L  must be a odd and LL   ,in form (2) the L  is also a odd but LL   ,in form (3) the L  is even and LL   .One can come t o a conclusion tha t there must be a al ternate chain in formed as (1) because ( 2.2). A alternate chain form ed as (1) is just a P -chain with resp ect to . However , this cont radicts our assumpt ion that there is no P- chain in G with r espect to . The theorem has be en proved. □ Follow s from above two theorem , one can draw conclus ion that: Basic Theorem The C hara cteristic nu mber of a Graph is a Graph inv ariant ,and a 2-factor in a graph G e xists if and only if th e Characteristic num be r of the graph i s zero. ♀ The graph G is decomposed into subgraph 12 , GG if 12 G G G  and 12 GG  =null graph. 3. Algorithm The Algorit hm for com puting maximum [0,2]- factor and characteristic number of gr aph. Theorem 2.1 and Theorem 2.2 shows that start from a initiatory [0,2]-factor 1 of G, if 1 ( ) 0 s T  we could to find a P- ch ain 1 L with respect to 1 then [0,2] -factor 2 1 1 L  have characterist ic num ber 21 ( ) ( ) 2 S S T T  Proceeding in this fashion , until a [0,2]-f actor k of G to be found an d there c an be no P-chain w ith respect to k , then k is t he m aximum [0,2]-f actor of G and ( ) S k T is the characteristic number of G.This is done by below Algorithm A. Algorithm A: To Co mpute a m aximum [0,2]-factor of G. Begin : input a null graph 0 of G . i : = 0 Do If Ts ( i )=0 then Output a 2-factor i of G ; End Else F ind a P - chain i L with respect to i . if t here i s not a P -chai n with respect to i then Output : A m aximal [0,2]-factor i and the char acteristi c number Ts( i ) of G.:end Els e 1 i i i L   i := i +1 End i f End if Loop End In A lgori thm A,We m ust t o find a P -chai n wit h resp ect a [0,2]- factor * of G, it can be complet ed by a DFS . That is ,from a vertex x with * ( ) 1 dx  to find a unmarked adjacent edges 12 ... i e e e alternately with respected to ( ) \ ( * ) E G E and ( * ) E , and mark them ,until a end -edge with a end-vertex y and * ( ) 1 dy  has been found , if for ji  there is no unmarked edge alternati ng with j e then back to 1 j e  ... . What is done by below Algor ithm . Algorithm B : To find o ut a P-chain with res pect a [0,2] -factor p of G. Begin: 1 ( ) p xV  and 1 ( ) 1 p dx  :1 i  Do If m o d 2 1 i  then If ther e is a unmarked edge 1 ( , ) ( ) \ ( ) i i i p e x x E G E   then i e to be mar ked I f 1 ( ) 1 p i dx   and 11 i xx   or 1 ( ) 0 p i dx   then Output a P-chai n: 12 ... i e e e : end Els e :1 ii  End if Elseif there i s no unmarked edge 1 ( , ) ( ) \ ( ) i i i p e x x E G E   then I f :1 i  then Output: there is no P -chai n from v er tex 1 x : end Els e :1 ii  End if End if Elseif m o d 2 0 i  then If ther e is a unmarked edge 1 ( , ) ( ) i i i p e x x E   then i e to be mar ked :1 ii  Elseif ther e is no unm arked edge 1 ( , ) ( ) i i i p e x x E   then :1 ii  End if Loop The Complexit y of A l gorithm The complexity of all of the algorithm is according to the complexi ty of the algorithm of finding a P -chai n with respect a [0,2]-factor p ,that is com plexity of algorithm B. The algorithm B based on DFS should to search a edge in () EG once only, so its complexity is o( | ( ) | EG ) ,therefor,the complexity all of the algorit hm is ( | ( ) | ) o n E G ,i.e 3 () on .We can get the below theorem immediately . Theorem 4.1 It can be decision in time 3 () on that if there is a 2 -fact or for a graph G, and get a 2-f actor when it exist. 4 Conclusion The chataterristic number of a graph, definited in this paper,is a graph invariant. It shows how far a graph can be [0,2]- factorizations and if and only if which is zero the graph exists 2-fact or.The given algor ithm m ake it can be decision and computed out in ti me 3 () on . H-cycl e is also a 2-factor,theref ore a necessary condition of that, a graph i s H-graph, is i ts characterist ic number is zero,and it also can be tested in 3 () on . The algori thms, gi ven in this paper,has be en programmed by VB . References [1] J. Akiyama and M. Kano. Book of Factors and Factorizations of Graphs . June, 2007. Online version: http://gorogoro.cis.ibaraki.ac.jp/web/papers/Factor GraphVer1A4.pdf [2] T. C. Biedl, P. Bose, E. D. Dema ine, and A. Lubiw. affecient algorithms f or Petersen's matching theorem. Journal of Algorithms , 38(1):110{134, 2001. [3] V. Chvatal. Linear Programming . W. H. Freeman, San Francisco CA, 1983. [4] P. Diaz-Gutierrez and M. Gopi. Quadrilateral and tetrahedral mesh stripi¯ cation using 2 -fac to r partitioning of the dual graph. The Visual Computer , 21(8-10):689{697, 2005. [5] M. Gopi and D. Eppstein. Single-strip triangulation of manifolds with arbitrary topology. Computer Graphics [6] C. Umans. An algorithm for fi nding Ham iltonian cycles in grid graphs without holes. Honors thesis , William s College, 1996. [7]Michael R.Garey and Dvid S.johnson Computers and intractability ,A Guide to the Theory o f NP -Completeness. P286,W.H Freeman and Company,San Francisco,1979.