ON STAR POLYNOMIALS , GRAPHICAL PARTITIONS AND RECONSTRUCTION

It is shown that the partition of a graph can be determined from its star polynomial and an algorithm is given for doing so. It is subsequently shown (as it is well known) that the partition of a graph is reconstructible from the set of node-deleted subgraphs.

edges.We define an m-S to be a tree consisting of a node of valency m m (called the centre of S joined to m other nodes.A 0-6 is a node and a m I-6Y is an edge.
Let G be a graph.A 6oY-COVe (or simply, a COVe) of G is a spanning sub- graph whose components are all stars.Let us associate with each m-star in G, an in- determinate or wig Wm+l; and wth each star cover C consisting of S where the summation is taken over all the star covers in G and (Wl, w 2 is a general weight vector.The basic results on star polynomials are given in the intro- ductory paper by Farrell [I]. We will show that HG the partition of a graph G can be obtained from E(G;).
This will then be used to show that R G is node-reconstructible, a result that can be established by more elementary means (see Tutte [2]).
For brevity, we will ite E(G) for E(G;w), since the same weight vector w will be used throughout the paper.Also, in partitions, we will use r k to denote the occur- of k r's.Finally, we will assume that () 0, for all r n. ence 2. STAR POLYNOMIALS AND GRAPHICAL PARTITIONS.
First of all, we state a lemma which can be easily proved.
LEMMA I. Let v be a node of valency d in G Then G contains () m-stars with centre v DEFINITION Let G be a graph with p nodes.A S%mpl m-covEr of G is a cover consisting of an m-star and p-m-1 isolated nodes.
It is clear that a simple m-cover in G will have weight w Wm+ in E(G).
A mon6mial of this form in E(G) will be referred to as a 6%mp Am, and its co- efficient, which will be denoted by c a 6dp e0e%eewt, c will be the number m m of simple m-covers in G.Note that the term w will also be a simple term.PROOF.This is straightforward.
The following lemma can be easily proved.
LEMMA 3. Let n be the largest valency of a node in G Then E(G) contains p-r-I all the terms w Wr+ (0 _< r _< n) with non-zero coefficient, i.e Cr # 0 for for (0 ! r in) Suppose that we put n in the above Lemma.Then G will consist of a set of disjoint edges and possibly isolated nodes.Clearly then the partition of G will be given by c p-2c H G (i 0 ). (2.1) Hence HG can be found from E(G).
If n I, then from Lemma 2, we get n r n r)br bk + E (k)br, for k > I. Ck rk (k   For n I, HG is given by Equation (2.1).For n 0, the result is trivial.
Theorem yields an algorithm for obtaining G from E(G).This algorithm is illustrated in the following example: EXAMPLE I. Let G be a graph such that E(G) First of all, we oserve from the term w6, that G has 6 nodes i.e. p 6.The sim- 7ww and w2w Therefore c 6 c =7 and c I.   Hence HG (312II)" It would be nice to be able to obtain G itself from E(G).From Theorem I, HG can be obtained.However there can be several graphs with the same partition.Since only the simple terms in E(G) are used to obtain G' it is not surprising that G itself is not clearly defined.Should G itself be clearly defined by the simple terms, then it would mean that the remaining terms of E(G) are useless as far as the characterization of G is concerned.It would be interesting to investigate the nature of these 'useless terms'.
Suppose that HG is unigraphic (i.e.there is only one graph with partition HG ), then G could be uniquely constructed from HG" Hence we have the following theorem.THEOREM 2. Let G be unigraphic.Then G can be constructed from E(G).
THEOREM 3. Let G and H be two graphs with p nodes.Then NG H if and only if E(G) and E(H) have the same simple coefficients.
PROOF.Suppose that the simple coefficients in E(G) and E(H) are equal.Then from Theorem I, G and H must have the same partition.Conversely, suppose that (r I) in E(G) and E(H) must NG H H Then from Lemma 2, the coefficients c r be equal.Finally, c   i, for all graphs.Hence E(G) and E(H) have the same simple coefficients.The result therefore follows.
3o STAR POLYNOMIALS AND RECONSTRUCTION.
The following theorem is analogous to the result for circuit polynomials given in Lemma 3 of Farrell and Grell [3], with i i.We suspect that the general result holds for all F-polynomials (see Farrell [4]).Here G-x denotes the graph obtained from G by removing node x.V(G) is the node set of G.
nl, j n2, j n It is clear that the monomial w w 2 ...w p'j is the weight of a cover with P one isolated node less than the corresponding cover in G.It is therefore the weight of a cover in G-x, for some node x in G. Hence it is a monomial of the polynomial Z E(G-x;).Conversely, every cover of G-x can be extended to a cover of G by adding an isolated node.Therefore every monomial m in Z E(G-x;) yields a corr- esponding monomial wlm in G.The derivative of wlm with respect to w yields E(G) a term with monomial m.It follows that w and Z E(G-x;w) have the same mono- mials.nl, j n2 n Since A. is the coefficient of w w 2 ,J...w p'j G has A covers con- 3 P j sisting of nl, j isolated nodes n2, j edges,..., n (p-l) stars.Suppose that P,j node x is removed from G. Then G-x will have a similar cover but with nl,j-I isolated nodes.Since node x could be any of the nl, j isolated nodes in the cover, it follows that each such cover in G gives rise to nl, covers with one less nl, j -I n 2 n isolated node.Hence the coefficient of w w 2 'j...w p'j in Z E(G-x;w) is p nl,jAj.From Equation (3.1) it follows that the monomials occur in Z E(G-x;) and E(G) with equal coefficients.Hence the result follows.
w Throughout the rest of this section, we will assume that the graph has at least three nodes.Also 'reconstructible' would mean node-reconstructible.By the deck D G a graph G we would mean the set {G-x: xeV(G)}.
It is well known (see Harary [5], Kelly [6], and Chatrand and Kronk [7]) that disconnected graphs are reconstructible.It follows that H G is reconstructible if G is disconnected, and therefore the number of isolated nodes in G is reconstruc- tible.We can however, prove the latter independently.The following lemma gives a connection between the number of isolated nodes and DG.
LEMMA 4. Let G be a graph with p nodes.Then G has r (>0) isolated nodes if and only if D G has exactly r graphs with r-I isolated nodes and p-r graphs with r or more isolated nodes.
PROOF.Suppose that G has r isolated nodes.Then D G must be of the form described.Conversely, suppose that D G is as described in the theorem.Let k be the number of isolated nodes in G. Since every element of D G has at least r-I isolated nodes, G cannot have less than r-l isolated nodes or it would mean that the removal of each node from G yields at least one new isolated node, and this is impossible unless G is a matching (in which case the result holds).Therefore k _> r Since D G contains no elements with less that r-I isolated nodes, k#r-l.
Hence k > r-l.But exactly r elements of D G has r-I isolated nodes and only one node can be removed at a time from G to form an element of D G. Therefore G has exactly r nodes each of whose removal reduces k to r-l.These nodes must be themselves isolated nodes.
k _> r.Clearly k r.Therefore k r and the result follows.
From the above lemma, we see that b 0 (of Theorem I) can be obtained from D G- is given, then we can find HG provided that all the re- maining br'S (0 < r < n) can be determined.The following result is well known (see Tutte [2]).We will give different derivation using star polynomials.THEOREM 5. H G is reconstructible.
PROOF.Let G be a graph with p nodes.From Theorem 4, we have, by inte- grating both sides with respect to w l, E(G) f(ZE(G-x;w)) + C(w 2,w Wp), where C(w2,w ,Wp) is a polynomial in the weights w2,w w p Suppose that D G is given.Then Z E(G-x;w) can be found.Hence / Z E(G-x;_ can be found.But this polynomial contains all the simple terms except w There-P fore all the simple coefficients of E(G), except c can be immediately found p-I We will consider two cases (i) Cp_2 0 and (ii) Cp_2 # 0.
If Cp_2 0, then from Lemma 3, Cp_ 0 Therefore all the simple coeffi- cients will be defined.It follows from Theorem I, that H G can be found.If Cp_2 # 0 then c will be unknown.However b will be known from D G (Lemma 4) p-i Therefore the system of equations in Theorem will have p-I equations and p-! un- knowns.It can be solved to find b I, b 2, bp_ I. Hence NG can be found.
In the proof of Theorem 5, we did not give any useful practical method for find- ing H G, when c 0. We shall do so now.p-2 The equations of Theorem can be written as follows:   w + 8ww2 + 19ww + 10WlW + C(w2,w3,w,w5).
[ poZyom%o of G (relative to the given weight assignment) is E(G;) Zw(C), is the number of edges in G. Therefore the sum of the valencies of the N n nodes of G is 2c kZ=l kb k b + kZ=2 kb k n =2c kE__2 kb k.
Let G be a graph with p nodes and let II G ple terms in E(G) are 6w w 2 Since the largest k for which w koccurs in E(G) is 4, it follows that n kb k 12 -2b 3b 12-8-3 I.
)bp_l + (p_2)bp_2 By adding and subtracting alternate equations, we get p-2c + c us denote the L.H.S. of this equation by S. Then S b + (-l)Pb p-l" bp_ (-I) p (S-D0)(3.3)SinceS can be found from the simple coefficients c (r p-2) and p, and b 0 r can be found (from Lemma 5), b can be found from Equation(3.3).Hence all the p-2 b's can be found by using the algorithm suggested by Theorem I.The following example illustrates the technique.

EXAMPLE 2 .
Figure I.It can be easily confirmed that2 2 By integrating with respect to w I, we get E(G)