PATH DECOMPOSITIONS OF CHAINS AND CIRCUITS

Expressions for the path polynomials (see Farrell [I]) of chains and circuits are derived. These polynomials are then used to deduce results about node disjoint path decompositions of chains and circuits. Some results are also given for decompositions in which specific paths must be used.

lent to shrink an incorporated subgraph to a "node".We will call such a "node" a compound node.A graph G containing an incorporated subgraph or a compound node i.e. a restricted graph, will normally be denoted by G*.
A tree with nodes of valencies i and 2 only will be called a chain.The chain with n nodes will be denoted by P By attaching a chain P to a nonempty graph G we n n will mean that a terminal node of P is identified with a node of G, so as to form a n connected component in which G and P are subgraphs with exactly one node in common.
n By adding a chain P to a graph G, we will mean that the two terminal nodes of P are n n attached to different nodes of G (N.B.In this case, G must have at least two nodes).
We refer the reader to Harary [2] for definitions of the standard terms used in graph theory.
Upper limits of summations will be infinity unless otherwise specified.A node disjoint path decomposition of G is another name for a path cover of G.The name is- land decomposition was used by Goodman and Hedetniemi [3], who derived an efficient algorithm for finding hc(T) the minimum number of edges needed to be added to a tree T in order to make it Hamiltonian.hc(G) is called the Hamiltonian completion number of G.It will be denoted by (G).Some results on (G) were obtained by Boesch, Cher and McHugh [4].Some results for trees were obtained by Slater [5].
In this paper we will use some of the basic results on path polynomials in order to derive the path polynomials of chains and circuits.From these polynomials, we will deduce results about node disjoint path decompositions of the graphs.Throughout the paper "decomposition" will mean "node disjoint path decomposition" and "cover" will mean "path cover".

BASIC PRELIIINARY RESULTS.
By putting the covers of G into classes according to whether or not they contain a specified edge, we obtain the following theorem, called the fundamental theorem for path polynomials.
THEOREM i.Let G be a graph and e an (unincorporated) edge of G. Let G" be the graph obtained from G by deleting e and G*, the graph obtained from G by path incor- porating e.Then P(G;w_) P(G';w_) + P(G*;w_).G (N.B Throughout this paper we will refer to graphs obtained by deleting edges as and graphs obtained by incorporating edges as G*, whenever Theorem I is applied.) The fundamental algorithm for path polynomials consists of repeated applications of Theorem i, until graphs Gi are obtained for which P(Gi;w)_ are known.The following lemmas imply some useful simplifications to the fundamental algorithm (also called the reduction process).In any practical application of the reduction process, we could incorporate an edge e, by identifying the nodes at the ends of e as described in Read [6] for chro- matic polynomials.However, we must keep track of the number of nodes in the incor- porated subgraph associated with the new node formed by identification.
Lemma 1 suggests that we can delete from G*, all edges which will finally complete circuits with incorporated edges.Also, if two edges have been incorporated at node x, then all unincorporated edges at x can be deleted.
The following result is called the Component Theorem.It is also given in [i] for general F-polynomials.It can be easily proved.
The notation G*[n] will be used for a restricted graph containing an incorporated chain with n edges.The following theorem gives a recurrence relation for the path polynomial of chains.
THEOREM 3. n P(Pn;W)-w i P(Pn_i;w)._i=l PROOF.Apply the reduction process to P by deleting a terminal edge.Then G' n will be a graph containing two components an isolated node and P G* will be n-l" P *[I].Apply the process to P *[I] by deleting the terminal edge adjacent to the in- n n corporated edge.In this case, the graph G' will be a graph with two components, an G* will be P * [2].Continue the re- incorporated edge having a weight w 2 and Pn-2" n duction process in this manner to subsequent graphs P *(k) (k > 2), until the entire n chain is incorporated.The result then follows by adding the contributions of the graphs G' formed during the process and the final incorporated graph P *[n-i], having n a weight w n For brevity, we will write P(n) for P(P ;w), when it would lead to no confusion.n Conventionally, we write P(0) I.The following table gives values of P(n) for n 1,2,3,4,5 and 6.Observe that the sum of the coefficients of the polynomial in th 2n-i the n row is The following theorem gives a generating function P(P ;w,t) for P(n).
n THEOREM 4. The result therefore follows.

P(Pn
We will now consider chains containing incorporated subchains.The covers of these restricted chains will be covers in which a particular path must appear.When the incorporated subchain contains r edges and contains a node of valency 1 of the restricted chain, the graph will be denoted by P *[r], where n is the number of nodes n in the nodes in the unincorporated subchain (including the node common to both sub- chains).The path polynomial of P *[r] is given in the following theorem.n THEOREM 5. n P(en*[r];w) Wr+i P(n-i).
i=l PROOF.Apply the reduction process to P *[r] by always deleting the edge which n links the restricted edges to the unrestricted ones.The result then follows.
When the incorporated subchain does not include a node of valency i of the re- P * [r] where n I and n 2 (nl,n 2 > i) are stricted chain, the graph will be denoted by nl+n2 the number of nodes in the unrestricted subchains.The path polynomial of this graph is given in the following result.THEOREM 6.
PROOF.Apply the reduction process to the graph, by deleting the edge of the Pnl will consist of subchain which is adjacent to an incorporated edge.The graph G I two components Pnl_l and P*n2[r].GI* will be P(n_l)+n2 [r+l].Apply the reduction process to GI*, and to subsequent Go*'Sl by deleting the edge of the unrestricted sub- chain of Pn. 1 which is adjacent to an incorporated edge.By adding the contributions of the G"s and the final restricted graph P * -l+r] we get  When the incorporated chain is attached to a node of valency 2 of the chain, the graph will be denoted by P * [r], where n I and n 2 (nl,n2 > i) are the number of nodes nl,n 2 in the subchains separated by the node of valency 3. The result for this graph is gi- ven in the following theorem.THEORY24 7.
i=l i=l PROOF.Let x be the node of valency 3 and y and z the nodes of the subchains P n 1 and Pn2 respectively, which are adjacent to x. Apply the reduction process by dele- ting the edge xy.G will be the graph consisting of components Pnl_ I and Pn[r].G* [r+l] Hence we get will consist of components Pn2_l and Pnl_l P(Pn,n2[r];w)-P(n I-i) P(Pn[r];w)_ + P(n 2-I) P(Pn_l[r+l];w)._ The result then follows by using Theorem 5.

S I{PLE PATH POLYNOMIALS OF CHAINS.
The factor (i +w) appears quite frequently in the simple path polynomials of chains and circuits.We will therefore replace it by z.Let us assume that i P(G;w) Aiz where the A.'s are constants.By putting w O, we get Z Ao 0, since P(G;w) can l i=l have no constant term.Also, the number of Hamiltonian paths in G (denoted by H(G)) will be the coefficient of w in P(G;w), which is 7 i A i. Hence we have i=l i LEMMA 2. Let P(G;w) We can obtain the simple path polynomial of P from Theorem 4 by putting w i w n for all i, then equating coefficients of tn.However, the result can be more easily obtained by a straightforward combinatorial argument.A cover of P with cardinality n k can be obtained by deleting k-1 edges for any 0 < k < n.Hence, we have THEOREM 8. n-i n n-i P(Pn;W) w(l+w) z z The following corollaries are immediate from Theorems 5 and 6.They can also be independently proved, by recognizing that in the case of the simple path polynomial of restricted chains, we can "shrink" the incorporated subchains to a node.
COROLLARY 5.1.For all r, n n-i P(Pn* [r];w) e(Pn;W) z z COROLLARY 6.1.For all r, nl+n2-1 nl+n2-2 P(P *.* [r];w) We can obtain the simple path polynomial of P * [r] by direct substitution into n I n 2 Theorem 7; however, this will be a bit tedious.It is better to obtain the result independently by using the reduction process.The result is given in the following corollary, for which an independent proof is given.nl,n2 PROOF.Apply the reduction process to the graph, just as we did in the proof of Theorem 7. G will consist of components P  G* will consist of components Pn2_l and Pnl_l* [r+l].But P(PnI_I* [r+l];w) P(Pnl_l;W).
Hence we get P (Pn ,n 2 [r];w) P(P i) P(n2) + P(n 2 I) P(n I i) The result then follows after simplification.k* Let us denote by Pn the restricted cha+/-n formed by attaching to a chain Pn' k incorporated paths (k < [n/2] such that no two are attached to adjacent nodes of P n We will also assume that none of the end-nodes of P are used, for if it is, then it n can be treated as an ordinary node, as implied by Corollary 5.1.The simple path k* polynomial of P is given in the following theorem.We can apply the reduction process to P (r+l)* by deleting the edge which is adjacent n to "last" incorporated path from the end of the restricted chain, such that G would consist of (i) a restricted chain containing r incorporated paths, and (ii) a restric- ted chain with one end-node attached to an incorporated path.Thus the components of r* G will be P and P * with s+t n Since the incorporated path is attached to an s t end-node of Pt*' P(Pt*;w) P(Pt;w).Hence s-2r-i r+l The graph G* will contain two components pr* and P Hence Hence the statement holds for k r+l, when it is assumed for k r.By the Princi- ple of Induction, it holds for all r.
(nl kl n2 k2 kr Let n be a partition of n.We will denote by NG() r the number of decompositions of the graph G into k paths with no nodes, for i 1,2,...,r.N G will denote the total number of decompositions of G into paths.It is clear that NG()_ is the coefficient of w k l n l w k2 w kr in P(G;w).The following corollary is immediate from Theorem 8.It confirms the observation made about the sum of the coefficients of the polynomials in The following corollary of Theorem 3 gives a recurrence which is useful for find- ing the number of decompositions of a chain into paths with specified lengths.NPn (nl n2 nr r) is the coefficient of Wnl Wn2 Wnr r in P(n).The only terms on the R.H.S. which contribute to this coefficient are w P(n-nl) n 1 Wn2P(n-n 2), and WnrP(n-nr )" The result therefore follows.
Corollary 3.1 suggests an algorithm for finding the number k 1 k 2 k r Np (n I n 2 nr The following lemma is quite useful when applying Coroln lary 3.1.Its proof is straightforward.I.gNNA 3.
By a restricted decomposition of a graph G, we mean a decomposition in which par- ticular paths must be included.Clearly, the decompositions of restricted graphs will be restricted.
The following corollaries can be easily deduced from the results given in Section 4 by putting w i.
COROLLARY 5.2.The number of restricted decompositions of the chain Pn+k in which a particular terminal subchain Pk is always included, is 2n-l.COROLLARY 6.2.The number of restricted decompositions of Pnl+n2+n3(nl,n2,n 3 > I) in which a particular subchain Pn3 (which does not include a terminal edge) is always nl+n2-2 included, is 2 where n I and n 2 are the numbers of nodes in the subchains se- parated by P n 3 COROLLARY 7.2.Let G be a graph consisting of 3 chains P ,P and P n I n 2 n 3 (nl,n2,n 3 > i) attached to a single node.The number of restricted decompositions of nl+n2-4 G, in which the chain P is always included, is 3.2 n 3 COROLLARY 9.1.Let G be the graph formed by attaching k chains to non-adjacent nodes of valency 2 in P The number of restricted decompositions of G, in which the n k chains are always included, is 3 k 2n-2k-l.
By extracting the coefficient of w s in the expression given for P(P * [r];w) in nl,n 2 k* Corollary 7.1 and the expression for P(Pn ;w) given in Theorem 9, we obtain the fol- lowing results.
COROLLARY 7.3.Let G be a graph consisting of 3 chains P P and P n I n 2 n 3 (nl,n2,n 3 > I) attached to a node.The number of restricted decompositions of G, with cardinality s(s > 1), in which the chain P is always included, is n 3 where N n I + n 2.
COROLLARY 9.2.Let G be a graph formed by attaching k chains to non-adjacent nodes of valency 2 of a Ldin P The number of restricted decompositions of G, with n cardinality s(s >k), in which the k chains are always included is 2 k-r r=0 k-r-1 6. PATH POLYNOMIALS OF CIRCUITS.
The notation C will be used for a circuit with n nodes.r=l PROOF.Apply the reduction process to C by deleting an edge The graph G" will n be P G* will be C *[i].Continue to apply the reduction process to the restricted n n graphs formed, by always deleting an edge which is adjacent to an incorporated edge.
The graphs G" will be restricted chains The graph formed by incorporating the final edge of Cn cannot contribute to P(Cn;W)_ since it contains an incorporated circuit.
By adding the contributions of the graphs G" formed during the process
n th

n
Observe that the sum of tile coefficients of the polynomial in the n row is i. w I + 6w14w2 + 6w13w3 + 9w12w2 + 6w12w4 + 12WlW2W3 + 6WlW5 + 2w23 + 6w2w4 + 3w 2 + 6w 6 A generating function for P(C ;w) can be obtained by using a technique similar to n that used in the proof of Theorem 4. It is given in the following theorem.
Let C* [r] be a restricted circuit consisting of an incorporated path with r n+r-i edges and a path with n nodes.Let us apply the reduction process to C* [r] as we n+ri did in the proof of Theorem ll for the restricted circuits.This will yield  If we apply the reduction process to this graph, n by deleting an unincorporated edge which is adjacent to an incorporated edge, the graph G" will be P+r_l[r-i] and G* will be P * [r].Hence we obtain the following q+r-I result by using Theorem 5. THEOREM 14.
n+r-i P(Cn**[rl];w) (Wr+ i + Wr+i_ I) P(n+r-i-i).In case (1), we can apply the reduction process to Ck*, n by deleting an edge which is adjacent to an incorporated chain and which belongs to a separating path of length greated than 2. G" will be P (k+l),, since the end-node to n which an incorporated chain is attached, can be treated as an ordinary node.G* will be Pn_l (k-1)*. 8. PATH DECOMPOSITIONS OF CIRCUITS.
The covers of P will also be covers of C Therefore, it is expected that the n n results similar to those given in Section 5 will also hold for circuits.
The following is an immediate corollary of Theorem 15.It confirms our observa- tion of the sum of the coefficients of the polynomials in Table 2. =2n-1 COROLLARY ii.i.N C n Since the covers of P will also be covers of C we have from Theorem i0, the n n following corollary.
LEMLA i. (i) If an incorporated edge creates an incorporated circuit in G*, then P(G*;w) 0. (ii) If an incorporated edge creates a node of valency 3 in G*, then P(G*;w__) 0.
nl+n2 s=lThe result follows by using Theorem 5.
ON k.Corollary 7.1 verifies the case for k= 1.Let us as- sume that the statement holds for k r.
i n-n i PROOF.From Theorem 3, we have P(n) w I P(n-l) + w 2 P(n-2) + + Wn.
i=O j =i Consider now the case in which the incorporated chain P * is attached to C We r n will denote this graph by C**[r-i].
i=l 7. SIMPLE PATH POLYNOMIALS OF CIRCUITS.The simple path polynomial of C can be easily deduced from Theorems 8 and ll, n or by elementary combinatorial argument.It is given in the following theorem.graph C*n+r_l[r] can be treated as Cn_ 1 by "shrinking" the incor- porated path to a node.Thus we have nresults for C**[r-1], we would solve the general problem in n n which k incorporated chains are attached to non-adjacent nodes of C (k<_ " ]).Let n us denote this graph by e ke n We will consider 2 cases: (i) C k* contains a pair of consecutive incorporated n chains which are separated by a path of length greater than 2, and (2) C k* has no such n pair of incorporated chains.
2), It follows thatIn case(2), G" will be Pn_l (k-* and G* will be Pnwe have the following theorem.THEOREM 16.If C k* contains a pair of consecutive incorporated chains, separated n by a path in C of length greater than 2, (or if k i

Table i
It can be observed from Table i, that all partitions of the integer n appear as n