ON THE NUMBER OF MINIMUM CUTS IN A GRAPH∗
L. SUNIL CHANDRAN† AND L. SHANKAR RAM‡
Abstract. We relate the number of minimum cuts in a weighted undirected graph with various structural parameters of the graph. In particular, we provide upper bounds for the number of minimum cuts in terms of the radius, diameter, minimum degree, maximum degree, chordality, girth, and some other parameters of the graph.
Key words. minimum cuts, circular partition AMS subject classifications. 05C35, 05C99 DOI.10.1137/S0895480103427138
1. Introduction. Let G = (V, E) be a graph or a multigraph with positive weights on its edges. In this paper, we will use n to denote|V|. By an unweighted graph, we mean that all the edges have unit weight. Let (A, A) denote a cut of G, defined by the subsets A ⊂ V and A =V −A. We denote by E(A, A) the set of edges in the cut, i.e.,E(A, A) ={(u, v)∈E:u∈Aand v∈A}. The weight of the cut (A, A) is defined as the sum of weights on all the edges in E(A, A), and will be denoted by w(A, A). A minimum cut (S, S) is one with the minimum weight over all cuts inG. (Some authors use the wordsglobal minimum cuts orconnectivity cuts instead of minimum cuts). We will denote the weight of the minimum cut in G by λ(G). Note that ifGis unweighted,λ(G) is the same as the edge connectivity of the graph, i.e., the minimum number of edges whose removal disconnects the graph.
Note that the minimum cut in a graph may not be unique. We use Λ(G) to denote the number of minimum cuts inG. The problem of counting the number of minimum cuts in a weighted undirected graph arises in various aspects of network reliability, like testing the super-λ-ness of a graph [8], estimating the probabilistic connectedness of a stochastic graph in which edges are subject to failure with probability p[4, 5, 6, 30], and other areas [29]. For example, for a sufficiently small p, the probabilistic connectedness ofGcan be approximated asP(G, p)≈1−Λ(G)pλ(G)(1−p)|E|−λ(G), suggesting the importance of counting and bounding Λ(G).
It is well known that for any weighted (positive weights) graph G, Λ(G)≤n
2
, and this upper bound is achieved ifGis a cycleCnofnnodes with each edge having weight λ(G)2 [12, 7, 22]. It is interesting to explore whether there exist tighter bounds for Λ(G) when the graph satisfies various properties. For example, Bixby [7] studies Λ(G) in terms of the weight of the minimum cutsλ(G) in the special case where all the edge weights are positive integers andλ(G) is an odd integer. For this case, Bixby [7]
shows that Λ(G)≤ 3n2 −2. In the case of unweighted simple graphs it is shown by Lehel, Maffray, and Preissmann [23] that ifλ(G) =k, wherek≥4 is an even positive
∗Received by the editors May 1, 2003; accepted for publication (in revised form) December 17, 2003; published electronically August 19, 2004. A preliminary version of this paper appeared in Proceedings of the 8th International Computing and Combinatorics Conference, Lecture Notes in Comput. Sci. 2387, Springer-Verlag, New York, 2002.
http://www.siam.org/journals/sidma/18-1/42713.html
†Max-Planck Institute for Informatik, Stuhlsatzenhausweg 85, 66123 Saarbr¨ucken, Germany ([email protected]). Most of this research was done while this author was in the Indian Institute of Science, Bangalore, and was supported in part by the Infosys Fellowship.
‡Department of Computer Science, Swiss Federal Institute of Technology, Haldeneggsteig 4, CH- 8092, Z¨urich, Switzerland ([email protected]).
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integer, then Λ(G)≤ (k+1)2n22 +(kk+1−1)n. Whenk >5 is an odd integer, they show that Λ(G)≤(1 +k+54 )n. The inherent structural difference between graphs with odd and even edge connectivity was pointed out by Kanevsky [21] also.
In this paper, we provide upper bounds for Λ(G) in terms of many other important parameters of graphs. We assume weighted graphs, unless otherwise specified. Multi- graphs, as far as the results here are concerned, can be considered as a special case of weighted graphs, since the multiedges can be replaced by a single edge of appropriate weight without affecting the value of Λ(G). Our only assumption about the weights is that they are positive. Note that, for the purposes of this paper, this assumption is equivalent to the assumption that the weights are at least 1, since multiplying the weights on every edge by the same constant will not change Λ(G). While our upper bounds are valid for weighted undirected graphs and multigraphs, in most cases, the properties in terms of which the upper bounds are stated depend only on the structure of those graphs. In other words, the radius or minimum degree in terms of which we describe the upper bounds are those of the underlying unweighted simple graph and do not depend on the weights of the edges.
There is an abundance of literature regarding the determination of λ(G) and finding a minimum cut in G. The problem of enumerating all the minimum cuts is considered by many authors [12, 28, 14, 15], and various data structures are invented to efficiently represent all the minimum cuts in a graph. (The currently fastest deter- ministic algorithm for computing all minimum cuts in a nonnegative, real weighted graph is due to Nagamochi, Nakamura, and Ishii [26].) The fact that the performance of some of these algorithms depends on the number of minimum cuts in the graph also makes it interesting to look for tighter upper bounds for Λ(G) whenGsatisfies certain properties. (For example, a randomized algorithm due to Karger builds a data structure that represents all minimum cuts inO(Λ(G) +nlogn) space.) See [14] for a brief survey of results regarding the enumeration of all minimum cuts.
The slightly different question of determining upper bounds for the number of approximate minimum cuts, i.e., those cuts having weight at most f λ(G), where f >1 is a constant, is considered in [31, 22, 27, 20]. For example, Karger [22], uses probabilistic analysis to show that there are at mostO(n2f) cuts of the above kind in a graph ofn nodes. Nagamochi, Nishimura, and Ibaraki [27] show that the number of cuts of weight at most 43λ(G) is bounded above byn
2
. Henzinger and Williamson [20] show an upper bound ofO(n2) for the number of cuts of weight at most 32λ(G), extending the arguments of [27].
1.1. Our results. Radius and diameter. If G = (V, E) is a connected graph, theeccentricity of a nodev ∈V is defined ase(v) = maxdistance(v, u) over all the nodesu∈V. We define the radius of the graphGasr(G) = minv∈Ve(v). A vertexv is a central node ife(v) =r(G). We define the diameter ofGasd(G) = maxv∈Ve(v).
(Note that, in this paper, by “distance” we mean only the distances in the underlying unweighted graph. Thus radius, eccentricity, and diameter have nothing to do with the weights.) We show that the number of minimum cuts Λ(G)≤(r+ 1)n−(2r+ 1)≤ (d+ 1)n−(2d+ 1), where Gis a weighted graph (positive weights) andr, dare the radius and diameter ofG. As a special case, we observe that if there is a node which is a neighbor of every other node in the graph, i.e., ifr(G) = 1, then Λ(G)≤2n−3.
We illustrate the tightness of this bound by constructing a weighted clique Kn for which Λ(Kn) = 2n−3.
Minimum and maximum degree. Let the minimum degree and maximum degree of Gbeδand ∆, respectively. (Note that minimum and maximum degrees have nothing
to do with the weights, i.e.,δ= minu∈V |N(u)|and ∆ = maxu∈V |N(u)|,N(u) being the set of neighbors of the nodeu). We show that Λ(G)≤(2(δ+1)3n + 1.5)n−(δ+13n + 2) and Λ(G)≤ (n−∆+3)n2 −(n−∆ + 2). Note that these bounds become significant when the involved parameters are reasonably large. Also it is easy to get an upper bound involving bothδand ∆, by extending the techniques discussed in the paper.
Chordality. Let C be a simple cycle of a weighted undirected graph G. Any edge in the induced subgraph on the nodes of C, G[C], other than the cycle edges themselves, is called a chord of C. C is called an induced cycle (or chordless cycle)1 if and only if C does not have any chords. The length of the largest induced cycle in a graphG is called chordality ofG. A graph G is calledk-chordal if and only if the chordality of G is at most k. We show that Λ(G)≤ (k+1)n2 −k, where k is the chordality of the underlying unweighted (simple) graph corresponding toG. We also show the tightness of the bound by exhibiting a k-chordal graph G for arbitrarily largensuch that Λ(G) =(k+1)n2 −k.
The word “chordality” originates from the well-known subclass of perfect graphs, the chordal graphs. A graph G is chordal if and only if there is no induced cycle of length 4 or more in G. We define the chordality of a chordal graph to be 3. All graphs other than chordal graphs have chordality≥4. Some other important classes of graphs with low chordality value are the cocomparability graphs, chordal bipartite graphs, and weakly chordal graphs, all of which are known to be 4-chordal. It can be easily shown that asteroidal triple-free (AT-free) graphs have chordality at most 5.
Thus, by substituting the appropriate values for chordality in the above upper bound, we obtain a list of results for various special classes of graphs.
Note that Cn (the cycle on n nodes) is the graph with maximum chordality amongst all graphs on n nodes. Also, it is a graph which contains the maximum number of minimum cuts possible, namely, n
2
= (n+1)n2 −n. (In fact, our bound given above shows thatCn with each edge having weight λ2 is theonly graph which containsn
2
minimum cuts, the weight of the minimum cut beingλ). The fact that the maximum value of Λ(G) is achieved by the graph of largest chordality motivates a study of the influence of chordality on Λ(G).
Girth. Girth is the length of the smallest cycle in G. We show that if G is an unweighted graph with girth g and minimum degree δ, then Λ(G) < (x+1n + 1)n− (x+12n + 1), wherexis an integer greater thane−2(2(δ−1)g−22 −2). Note that this is in contrast with the bound in terms of chordality, the length of the largest induced cycle.
The Fiedler value. The Laplacian matrix of a graphGis defined asL=D−A, whereAis the adjacency matrix andD is the diagonal matrix whose (i, i)th entry is the degree of theith vertex in G. The smallest eigenvalue ofL can be shown to be equal to 0. The second smallest eigenvalueµofLis sometimes known as the Fiedler value ofG. This is a well-studied graph parameter. It can be easily shown that ifGis a regular graph, thenµis equal to the gap between the two highest eigenvalues of the adjacency matrix A ofG. Various structural parameters of a graph (like diameter, vertex connectivity, vertex and edge expansion, and bisection width) are known to be related toµand in general to the eigenvalues ofA orL. (See [13, 3, 25, 1].)
We observe that ifµis above the threshold value 1+n−δδ, whereδis the minimum
1An induced cycle or a chordless cycle is often called a “hole” in the perfect graph literature.
Recall that the strong perfect graphtheoremcharacterizes perfect graphs in terms of odd holes and antiholes.
degree, then all the minimum cuts in an unweighted graph Gare single vertex cuts.
In general, ifµis the Fiedler value andλis the edge connectivity ofG, we show that Λ(G)≤(2λµ2+3)n−(2λµ+ 2) provided2λµ<n3.
2. Preliminaries. Consider an undirected graphG= (V, E) with a weight func- tionw: E→ +. Let U and W be disjoint subsets of V. LetE(U, W) ={(u, v)∈ E : u∈ U, v ∈ W} be the set of edges between the vertices in U and the vertices in W. Also, let w(U, W) be the sum of the weights on the edges in E(U, W). As mentioned in the introduction,λ(G) denotes the weight of a minimum cut, and Λ(G) denotes the number of minimum cuts inG. LetX⊂V. We will denote the induced subgraph onX byG[X].
Lemma 2.1. If (S, S)is a minimum cut of a connected undirected graphG, then G[S]andG[S] are connected.
Proof. Suppose thatG[S] is not connected. LetG[S1] be a connected component ofG[S], whereS1⊂S. Clearly (S1, S1) is a cut of Gandw(S1, S1)< w(S, S) since E(S1, S1) ⊂ E(S, S). But this is a contradiction since (S, S) is assumed to be a minimum cut.
Definition 2.2. Let (X, X) and (Y, Y) be two cuts in a weighted undirected graph. (X, X)and (Y, Y) are said to cross each other if and only if all the four sets X∩Y,X∩Y,X∩Y, and X∩Y are nonempty. Then (X, X)and (Y, Y)are called a crossing pair of cuts.
Lemma 2.3. A pair of cuts (S, S) and (P, P) do not cross if and only if S (or S) is a subset ofP or P. (That is,S ⊆P,S⊆P,S⊆P, or S⊆P.)
Proof. The proof follows from the definition of a crossing pair of cuts.
Lemma 2.4 (Bixby [7] and Dinic, Karzanov, and Lomosonov [12]). Let (X, X) and (Y, Y)be a crossing pair of minimum cuts in a weighted undirected graphG. Let A=X∩Y,B=X∩Y,C=X∩Y, andD=X∩Y. Then,
1. w(A, B) =w(B, D) =w(D, C) =w(C, A) = λ(G)2 ;
2. w(A, D) =w(B, C) = 0. That is,E(A, D)∪E(B, C) =∅.
Lemma 2.5. If (P, P) and (S, S) are a crossing pair of minimum cuts, then E(P, P)∩E(S, S) =∅.
Proof. E(P, P)∩E(S, S) = E(S∩P, S∩P)∪E(S ∩P, S∩P) = ∅ by Lem- ma 2.4.
Definition 2.6. A circular partitionC = (U0, U1, U2, . . . , Uk−1) (where k≥4) of the vertices of a graph G is a partition of the set of vertices V of G into disjoint nonempty subsetsU0, U1, . . . , Uk−1 such that the following hold:
1. w(Ui, Ui+1 mod k) = λ(G)2 for 0≤i≤k−1.
2. Ifi=j+ 1mod k ori=j−1mod k, thenw(Ui, Uj) = 0; i.e.,E(Ui, Uj) =∅. 3. For 0≤i≤k−1, the cut (Ui, Ui)—which is a minimum cut by conditions
1 and 2—does not cross with any other minimum cut (A, A)inG.
Definition 2.7. A cut (A, A) is called a union cut with respect to a circular partitionC= (U0, U1, . . . , Uk−1)if and only if there exists somei,0≤i≤k−1, such that A=i+b−1mod k
j=i Uj, where 2≤b≤k−2. (Note that bothA and A contain at least two subsets inC). The cut (A, A)is called a subset cut with respect to C if and only ifA⊆Ui or A⊆Ui for somei.
Lemma 2.8. Let C = (U0, U1, . . . , Uk−1) be a circular partition ofG. Then any minimum cut (S, S) of G is either a union cut or a subset cut with respect to C. Moreover, every union cut with respect to C is a minimum cut inG.
Proof. By the definition of a circular partition, (S, S) does not cross with any of the minimum cuts (Ui, Ui). Therefore, by Lemma 2.3 S ⊆Ui, S ⊆Ui, S ⊆Ui, or S ⊆Ui. Suppose (S, S) is not a subset cut. Then, we haveUi ⊆S or Ui⊆S for all i. Since by Lemma 2.1,G[S] and G[S] are connected, we infer that (S, S) is a union cut. Also, that a union cut is a minimum cut follows easily from conditions 1 and 2 of the definition of a circular partition (Definition 2.6).
Lemma 2.9. Let G be a weighted undirected graph. Then G has a circular partition C = (U0, U1, . . . , Uk−1), wherek ≥4, if and only if there exists a crossing pair of minimum cuts in G.
Proof. If there is a circular partitionC= (U0, U1, . . . , Uk−1) withk≥4, clearly the minimum cuts (U0
U1, U0
U1) and (U1
U2, U1
U2) cross with each other. On the other hand, if there is a crossing pair of minimum cuts inG, namely, (S1, S1) and (S2, S2), due to a theorem of Bixby [7] and Dinic, Karzanov, and Lomosonov [12], there exists a circular partition C = (U0, U1, . . . , Uk−1) such that each of S1∩S2, S1∩S2,S1∩S2, andS1∩S2equalsb−1
i=aUi for appropriate choices foraandb. The
“if” part of the Lemma follows immediately from this.
For a circular partition C of G, let thepartition number p(C) be defined as the number of subsets inC. We define the partition number of the graphGas follows.
Definition 2.10. The partition numberp(G)of a graphGis defined asp(G) = 3 if there is no circular partition for G. Otherwise, p(G) = maxp(C), over all circular partitionsC of G.
Note that if there is a crossing pair of minimum cuts in G, thenp(G) ≥ 4, by Lemma 2.9. Otherwise,p(G) = 3.
Definition 2.11. By contraction of a subset of vertices X ⊂ V, we mean re- placing all the vertices inX by a single vertexxand adding the edges (y, x)for each y ∈N(X), where N(X)is the set of neighbors2 of X. The weight of the edge (y, x) (where y ∈ N(X)) is assigned to be w(y, x) =
z∈Xw(y, z), where (y, z) ∈ E(G).
We denote the graph obtained after the contraction operation by G/X. We will refer to the operation of undoing the effect of a contraction (i.e., restoring G from G/X) by putting backX in the place ofx, as expanding the nodex.
Lemma 2.12. If (S, S) is a minimum cut in a weighted undirected graph G such that no other minimum cut (A, A) crosses with (S, S), then Λ(G) = Λ(G/S) + Λ(G/S)−1.
Proof. Note that since (S, S) is a minimum cut, the value of the minimum cut in G/S and G/S will be the same as that inG. First we claim that Λ(G)≤Λ(G/S) + Λ(G/S)−1. This can be seen by observing that corresponding to each minimum cut inGthere is a minimum cut in eitherG/SorG/S. This follows from the assumption that no minimum cut (A, A) of Gcrosses with (S, S) and so exactly one of the four casesA⊃S,A⊃S,A⊃S, orA⊃S is true by Lemma 2.3. Thus the minimum cut (A, A) remains intact either inG/S or G/S. Also, (S, S) appears in both G/S and G/S, which accounts for subtracting 1. To see Λ(G)≥Λ(G/S) + Λ(G/S)−1, observe that any minimum cut (A, A) inG/S or inG/S has a corresponding minimum cut in G. For example, consider a minimum cut (A, A) inG/S. Without loss of generality, let the nodesinG/S (which corresponds to the contraction ofS) be inA. When we expand s, clearly the minimum cut (A
S− {s}, A
S− {s}) of Gcorresponds to the minimum cut (A, A) ofG/S. Moreover, it can be easily verified that the cuts ofG which correspond to the cuts ofG/Sare distinct from the cuts ofGwhich correspond
2N(X) ={u∈V −X: There exists a nodev∈Xsuch that (u, v)∈E}.
to the cuts ofG/S except for (S, S), which is accounted for by subtracting 1. Hence the result follows.
Lemma 2.13. If there are no crossing pairs of minimum cuts inG, thenΛ(G)≤ 2n−3. Moreover, there exists a graph on n nodes, Gn (for everyn≥2), such that Λ(Gn) = 2n−3.
Proof. If n= 2, clearly Λ(G) = 1, and the lemma is true. Assume that for all graphs with number of nodes< n(wheren≥3), the lemma is true. Consider a graph G on n nodes with no crossing pairs of minimum cuts. If all the minimum cuts of Gare singlenode cuts (i.e., of the form ({u},{u})), then clearly there are at most n minimum cuts. Then, Λ(G)≤n≤2n−3. Otherwise, there is a minimum cut (S, S) such that|S| ≥2 and|S| ≥2. LetG1=G/S andG2=G/S. Also, let the number of nodes inG1and G2 be n1 andn2, respectively. Since any minimum cut (A, A) ofG does not cross with (S, S), by Lemma 2.12 we have Λ(G) = Λ(G1) + Λ(G2)−1. Also, it can be easily verified that there will not be any crossing pair of minimum cuts in G1orG2, since such a pair will give rise to a corresponding pair of crossing minimum cuts inGalso, which is a contradiction. Thus sinceG1 andG2 have< nvertices, we have Λ(G)≤2n1−3 + 2n2−3−1 = 2(n+ 2)−7 = 2n−3, sincen1+n2−2 =n. In Theorem 5.2, we show a way to assign weights to the edges of a cliqueKn such that Λ(Kn) = 2n−3, illustrating the tightness of the bound given by this Lemma.
3. Partition number,p(G).
Lemma 3.1. Let Gbe a weighted undirected graph. If (X, X)is a minimum cut of Gsuch that no other minimum cut crosses with (X, X), thenp(G/X)≤p(G).
Proof. Suppose p(G/X) = p > p(G) = p. Clearly p ≥ 4 (since by def- inition of the partition number, p(G) = p ≥ 3). Consider a circular partition C = (U0, U1, . . . , Up−1) of G/X. Without loss of generality, assume that the node xobtained by contracting X is present in U0. We claim thatC = (W0, . . . , Wp−1), whereW0= (U0− {x})∪X andWi=Uifor 0< i≤p−1, is a circular partition for G. (This will clearly contradict the assumption that p(G/X) =p > p(G), proving the lemma).
Suppose C = (W0, W1, . . . , Wp−1) is not a circular partition for G. Then, by definition of circular partition, there exists a minimum cut (A, A) ofGwhich crosses with (Wi, Wi) for somei.
Case 1. (A, A) does not cross with (W0, W0) but it crosses with (Wi, Wi) for some i >0.Since (A, A) does not cross with (W0, W0) by Lemma 2.3, we have (1)A⊆W0, (2)A⊆W0, (3)A⊆W0, or (4)A⊆W0.
Case 1.1. A ⊆ W0 or A ⊆ W0. Since W0 ⊆ Wi, we have A ⊆ Wi or A⊆Wi, respectively. So, in both cases (A, A) does not cross with (Wi, Wi) by Lemma 2.3, contradicting the assumption of Case 1.
Case1.2. A⊆W0 or A⊆W0, i.e.,W0⊆Aor W0⊆A, respectively. Since X ⊆W0,X ⊆AorX ⊆A, which means that all the nodes inX are on the same side of the cut (A, A). Thus, the cut (A, A) ofG/X corresponding to (A, A) is a minimum cut of G/X. Since X ⊆ W0 ⊆ Wi, X ⊆Wi∩A or X ⊆Wi∩A; i.e., all the nodes in X are either inWi∩A orWi∩A. But since (Wi, Wi) crosses with (A, A), the sets Wi∩A, Wi∩A, Wi∩A, and Wi∩Aare nonempty. Therefore, clearly when we contractXto getG/X, the corresponding four setsUi∩A,Ui∩A,Ui∩A, andUi∩Aare also nonempty.
This means that in G/X, (A, A) crosses with (Ui, Ui), contradicting the assumption thatC = (U0, U1, . . . , Up−1) is a circular partition forG/X.
Case2. (A, A)crosses with(W0, W0). Remember that by assumption (A, A) does not
cross with (X, X). We have the following four possibilities by Lemma 2.3: (1)A⊆X, (2)A⊆X, (3)A⊆X, or (4)A⊆X.
Case 2.1. A⊆X or A⊆X. SinceX ⊆W0, we have A⊆W0 or A⊆W0, respectively, which means that by Lemma 2.3, (A, A) does not cross with (W0, W0) in both cases, contradicting the assumption of Case 2.
Case 2.2. A ⊆ X or A ⊆ X; i.e., X ⊆ A or X ⊆ A, which means that all the nodes in X are on the same side of (A, A). Thus, the cut (A, A) of G/X corresponding to (A, A) is a minimum cut of G/X. Since X⊆W0, we have X ⊆W0∩A orX ⊆W0∩A, i.e., all the nodes inX are completely in W0∩AorW0∩A. But since (W0, W0) crosses with (A, A), the setsW0∩A, W0∩A, W0∩A, and W0∩A are nonempty. Therefore, clearly when we contractX to getG/X, the corresponding four setsU0∩A,U0∩A,U0∩A, and U0∩A also are nonempty. This means that in G/X, (A, A) crosses with (U0, U0), contradicting the assumption that C = (U0, U1, . . . , Up−1) is a circular partition forG/X.
Thus, we infer that no minimum cut (A, A) can cross with any cut (Wi, Wi) in the circular partition C = (W0, W1, . . . , Wp−1) for G. But p(C) =p > p=p(G), a contradiction. Thus we havep(G/X)≤p(G).
In the following lemma, we provide an upper bound for Λ(G) in terms of the partition number. The tightness of the lemma will be established in Theorem 9.1.
Lemma 3.2. Let G= (V, E) be a weighted undirected graph, where |V|=n≥2, and let the partition numberp(G)≤p. Then,Λ(G)≤(p+1)n2 −p.
Proof. The proof is by induction on n. If n = 2, by definition p(G) = 3, and it is easy to verify that Λ(G) ≤ (p+1)n2 −p. Now, assume that for all graphs with number of nodes< n(wheren≥3), the lemma is true. LetGbe a graph onnnodes (n≥3) withp(G) =p. Ifp= 3, then by Lemma 2.9 there are no crossing pairs of minimum cuts inG, and hence by Lemma 2.13 Λ(G)≤2n−3 = (p+1)n2 −p. Now, let p≥4. By Lemma 2.9 there exists a circular partitionC= (U0, U1, . . . , Up−1) ofG. If p=n, then|Ui|= 1 for eachiand clearlyG=Cn, a cycle ofnnodes with each edge having weight λ(G)2 . Therefore, it hasn
2
= n(n2−1) = (n+1)n2 −nminimum cuts. If p < n, then there exists a Ui∈ C such that|Ui| ≥2. LetG1 = (V1, E1) =G/Ui and G2= (V2, E2) =G/Uibe the graphs obtained by contractingUiandUi, respectively.
Since|Ui| ≥2 and|Ui| ≥3 (note that this follows fromp(G)≥4), clearlyn >|V1| ≥4 and n > |V2| ≥ 3. Let p1 =p(G1) andp2 = p(G2). Note that by the definition of a circular partition, no minimum cut (A, A) of G crosses with (Ui, Ui). Hence by Lemma 2.12 we have Λ(G) = Λ(G1) + Λ(G2)−1. Now, by the induction assumption, Λ(Gj)≤ (pj+1)n2 j −pj (forj = 1,2) and we have
Λ(G)≤ (p1+ 1)n1
2 −p1+(p2+ 1)n2
2 −p2−1.
(3.1)
By Lemma 3.1 we havep1=p(G/Ui)≤pandp2=p(G/Ui)≤psince the minimum cut (Ui, Ui) does not cross with any other minimum cut inG. Forni≥2 andpi≤p (i= 1,2), it is easy to verify that (pi+1)n2 i−pi≤ (p+1)n2 i−p. Substituting in inequality (3.1) and noting thatn1+n2−2 =n, we get Λ(G)≤(p+1)n2 −p.
In the rest of the paper, we show that various structural parameters of a graph can influence the partition number p(G). Thus by means of Lemma 3.2 we relate the number of minimum cuts, Λ(G), with many seemingly unrelated properties of the graph.
Remark. Please note that if n≥2 andx≥p, then (x+1)n2 −x≥ (p+1)n2 −p. In most of the theorems below, we show thatpis bounded above by a function f(y) of y, wherey is some parameter ofG, thereby showing that Λ(G)≤ (f(y)+1)n2 −f(y).
4. Radius and diameter. IfG= (V, E) is a connected graph, theeccentricity of a node v ∈ V is defined as e(v) = maxdistance(v, u) over all the nodes u∈ V. We define the radius of the graphG asr(G) = minv∈V e(v). A vertexv is a central node ife(v) =r(G). We define the diameter ofGasd(G) = maxv∈V e(v). (Note that by “distance” we mean only the distances in the underlying unweighted graph. Thus radius, eccentricity, and diameter have nothing to do with the weights.)
Theorem 4.1. If ris the radius of a weighted undirected graph G, thenΛ(G)≤ (r+ 1)n−(2r+ 1)(wheren≥2).
Proof. Suppose there are no crossing pairs of minimum cuts in G. It follows by Lemma 2.13 that Λ(G)≤2n−3. Since the radius is at least 1, it is easy to verify that Λ(G)≤(r+ 1)n−(2r+ 1) in this case. Otherwise, by Lemma 2.9 there exists a circular partitionC = (U0, U1, . . . , Up−1) forG, where p= p(G)≥4. Let x∈Ui
be a central node ofG. Lety∈Ui+p
2mod p. Clearlydistance(x, y)≥ p2. That is, r≥ p2orp≤2r+ 1. Now, by Lemma 3.2 we get Λ(G)≤(r+ 1)n−(2r+ 1).
We note that the bound given by the above theorem can be tight. For example, considerC2n+1, the cycle on 2n+ 1 nodes. Clearly the radius ofC2n+1 isn, and the number of minimum cuts =2n+1
2
= (n+ 1)(2n+ 1)−(2n+ 1).
Observe that similar arguments as given for the case of the radius also hold well for the diameter. Thus,
Λ(G)≤(d+ 1)n−(2d+ 1).
This can also be verified from Λ(G)≤(r+ 1)n−(2r+ 1)≤(d+ 1)n−(2d+ 1) by noting thatd≥randn≥2.
5. Universal node. An interesting special case of Theorem 4.1 occurs when radius(G) = 1. Then, there exists a node which is adjacent to every other node of the graph. (Such a node is called auniversal node.) Thus, if there is a universal node in the graph, then Λ(G)≤2n−3 by Theorem 4.1. In fact, a stronger statement is true.
Theorem 5.1. If there is a universal node u in G, then there cannot be any crossing pairs of minimum cuts inG.
Proof. If there is a crossing pair of minimum cuts, then by Lemma 2.9 there is a circular partition C = (U0, U1, . . . , Uk−1) (k ≥ 4). Without loss of generality let u∈U0. Clearlyucannot be adjacent to any node inU2, by the definition of circular partition, contradicting the assumption thatuis a universal node.
Note that in a complete graph,Kn, every node is a universal node. Thus, there are no crossing pairs of minimum cuts in a clique. Below, we show a way to assign weights to the edges ofKn such that the number of minimum cuts Λ(Kn) = 2n−3, thus illustrating that the bound of Lemma 2.13 is tight. Moreover, since the radius of a clique is 1, this is a tight example for Theorem 4.1 too. Since a complete graph is a chordal graph, the example below also illustrates the tightness of Theorem 7.2.
Theorem 5.2. For any n≥2 andλ >0, there exists a weighted complete graph Kn such thatλ(Kn) =λandΛ(Kn) = 2n−3. Moreover, every node xofKn defines a minimum cut ({x},{x})ofKn.
Proof. For n = 2, it is trivial. For n = 3, let K3 be the triangle with each edge having weight λ2. Clearly Λ(K3) = 2·3−3 = 3. Also, note that every node x
in K3 defines a minimum cut ({x},{x}). Now inductively assume that there exists a weighted complete graph on n−1 nodes Kn−1 (n ≥ 4) such that λ(Kn−1) = λ, Λ(Kn−1) = 2(n−1)−3 = 2n−5, and every node x of Kn−1 defines a minimum cut ({x},{x}) in Kn−1. We show how to construct a weighted complete graph Kn
from Kn−1 such thatλ(Kn) =λ, Λ(Kn) = 2n−3, and every node xofKn defines a minimum cut ({x},{x}) inKn.
Let u be any node of Kn−1. We remove u from Kn−1 (along with the edges incident on it) and then add two other nodes u, u in its place. From each node y in Kn−1 (y = u), we add the edges (y, u) as well as (y, u) to the new nodes and assign weights w(y, u) =w(y, u) = w(y,u)2 . We also add the edge (u, u) with w(u, u) = λ2.
It is easy to see that the new graph Kn is a complete graph. Let S ={u, u}. Since ({u},{u}) is a minimum cut of Kn−1, w(S, S) = λ(Kn−1) = λ. We claim that λ(Kn) = λ. If not, there exists a cut (A, A) in Kn such that w(A, A) < λ. If both the nodes ofS ={u, u} are present on the same side of the cut (A, A), then the corresponding cut inKn−1 obtained by contracting S will also have weight< λ, contradicting the assumption thatλ(Kn−1) =λ. Therefore, without loss of generality, we can assume that u ∈A and u ∈A. Then clearly (u, u)∈E(A, A). Also, for eachyinKn(y=u andy=u), exactly one of the edges (u, y) or (u, y) belongs to E(A, A). Now recall thatw(y, u) =w(y, u) =w(y,u)2 and
y=uw(y, u) =λinKn−1. So we have w(A, A) ≥ w(u, u) + 12
y=u,u(w(y, u) +w(y, u)) = λ2 + λ2 = λ, which is a contradiction to the assumption that w(A, A) < λ. Hence, λ(Kn) = λ and (S, S) is a minimum cut of Kn. Also, since Kn is a clique, by Theorem 5.1 no minimum cut (A, A) of Kn crosses with (S, S). By Lemma 2.12 we have Λ(Kn) = Λ(Kn/S) + Λ(Kn/S)−1. But Kn/S = Kn−1 and Kn/S = K3. Thus we have Λ(Kn) = 2n−5 + 3−1 = 2n−3. Also it is easy to verify that every node xofKn
defines a minimum cut ({x},{x}) ofKn.
6. Maximum and minimum degree. The maximum degree ∆(G) (when it is reasonably high) can also constrain the number of minimum cuts Λ(G).
Theorem 6.1. If ∆ is the maximum degree of a weighted undirected graph G, thenΛ(G)≤ (n−∆+3)n2 −(n−∆ + 2), wheren≥2.
Proof. Suppose there are no crossing pairs of minimum cuts inG; then by Lem- ma 2.13, Λ(G) ≤ 2n−3 ≤ (n−∆+3)n2 −(n−∆ + 2), which will be true if 0 ≤ n2−(∆ + 3)n+ (2∆ + 2) or 0≤(n−∆−1)(n−2), which is true since n≥2 and
∆≤n−1. Now, if there is a crossing pair of minimum cuts inG, then by Lemma 2.9 there is a circular partition C = (U0, U1, . . . , Up−1) (p=p(G)≥4). Without loss of generality, let the maximum degree nodeu∈U1. Then,|U0∪U1∪U2| ≥∆ + 1 since every neighbor ofumust be inU0,U1, orU2. Thus,p≤3 + (n−∆−1) =n−∆ + 2 since eachUi of the circular partition must contain at least one node. By Lemma 3.2, Λ(G)≤(n−∆+3)n2 −(n−∆ + 2).
Interestingly, the minimum degree of the graph can also control the number of minimum cuts.
Theorem 6.2. If δ is the minimum degree of a weighted undirected graph G, thenΛ(G)≤(2(δ+1)3n + 1.5)n−(δ+13n + 2), wheren≥2.
Proof. If there are no crossing pairs of minimum cuts inG, it can be easily verified that Λ(G)≤2n−3≤(2(δ+1)3n + 1.5)n−(δ+13n + 2) for n≥2. Otherwise consider a circular partitionC= (U0, U1, . . . , Up−1) (p=p(G)≥4). Group the subsets inCinto p3triplets (U3i, U3i+1, U3i+2) for 0≤i≤ p3−1. |U3i|+|U3i+1|+|U3i+2| ≥δ+1 since
each neighbor of a nodeu∈U3i+1must be in one of the three sets in the corresponding triplet. Thus,p3(δ+ 1)≤n, and the result follows by Lemma 3.2.
7. Chordality. Recall that the chordality of a graph is the length of the longest induced cycle in the graph. We provide an upper bound for Λ(G) in terms of chordality in the following theorem. Its tightness is established in Theorem 9.1.
Theorem 7.1. IfGis a weighted undirected graph with chordalityk, thenΛ(G)≤
(k+1)n
2 −k, wheren≥2.
Proof. If there are no crossing pairs of minimum cuts inG, then by Lemma 2.13, Λ(G)≤2n−3≤ (k+1)n2 −k, sincekis at least 3 by definition andn≥2. Otherwise, consider a circular partitionCforGsuch thatp(C) =p(G). Ifp(C)> k, clearly there is an induced cycle in G with length > k, contradicting the k-chordality of G. It follows thatp(G)≤k. Therefore, by Lemma 3.2, Λ(G)≤ (k+1)n2 −k.
Theorem 7.2. If G is a weighted chordal graph, then Λ(G) ≤ 2n−3, where n ≥ 2. Moreover, there are no crossing pairs of minimum cuts in G. Also, there exists a weighted chordal graph G, for every n≥2, such that Λ(G) = 2n−3.
Proof. Since for chordal graphsk= 3 (by definition), Λ(G)≤2n−3 follows from Theorem 7.1. If there is a crossing pair of minimum cuts inG, then there is a circular partitionC forGwithp(C)≥4 by Lemma 2.9. This immediately implies an induced cycle of length≥4, contradicting the fact thatGis chordal. Finally, since complete graphs are chordal graphs, the construction of Theorem 5.2 establishes the tightness of this bound.
There are some interesting special classes of graphs which can be shown to have low chordality value. We list below a few results which immeditately follow from Theorem 7.1.
Cocomparability graphs consist of graphs whose complements are comparability graphs. See [17] for the definition of a comparability graph. It can be shown that the chordality of cocomparability graphs is at most four; see, for example, [16]. Thus by Theorem 7.1 we have the following theorem.
Theorem 7.3. If Gis a cocomparability graph on n vertices with positive edge weights,Λ(G)≤2.5n−4.
The class of weakly chordal graphs was introduced by Hayward in [19]. G is defined as a weakly chordal graph if and only if neither Gnor the complement ofG contains a chordless cycle of length at least 5. It follows from this definition that the chordality of weakly chordal graphs is at most 4. The class of weakly chordal graphs is quite a large one, as it contains the classes of cochordal graphs, chordal bipartite graphs, permutation graphs, trapezoid graphs, tolerance graphs, 2-threshold graphs, and others. Applying Theorem 7.1, we have the following result.
Theorem 7.4. IfGis a weakly chordal graph onnvertices with positive weights on its edges, thenΛ(G)≤2.5n−4.
An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is AT-free if it contains no asteroidal triples. AT-free graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs.
Theorem 7.5. If Gis an AT-free graph onnnodes with positive weights on the edges, then Λ(G)≤3n−5.
Proof. In view of Theorem 7.1, we just have to show that an AT-free graph doesn’t contain a chordless cycle of length at least 6. Suppose it contains a chordless cycle of length 6 or more. Then clearly we can pick three points from this cycle such that they form an independent set and any two of them has a path between them, which avoids
the neighborhood of the third. But this is not possible since the graph is assumed to be AT-free.
In fact there are many more special classes of graphs with low chordality value.
The interested reader is referred to [10].
8. The stability number. The stability numberαis defined as the size of the maximum independent set in the graph.
Theorem 8.1.
Λ(G)≤(α+ 1)n−(2α+ 1).
Proof. Sinceα is at least 1, if there is no crossing pair of minimum cuts in G, the theorem is clearly true. Otherwise it is easy to see that the partition number p≤2α+ 1.
9. A tight construction. We establish the tightness of Theorem 7.1 and Lem- ma 3.2 by the following construction.
Theorem 9.1. For each k ≥3 and λ >0, there exists an infinite family G of weighted undirectedk-chordal graphs such that for each graphGn∈ G withnnodes (n being an integer of the formk+q(k−2)forq= 0,1, . . .),Λ(Gn) = (k+1)n2 −k, weight of the minimum cut =λ, and p(Gn) =k. Moreover, every node uof Gn defines a minimum cut ({u},{u}).
Proof. When k = 3, the family of cliques constructed in Theorem 5.2 has the desired properties; i.e.,λ(Kn) =λ, Λ(Kn) = 2n−3,p(Kn) = 3, and every node xof Kn defines a minimum cut ({x},{x}). In the rest of the proof we assume thatk≥4.
First note thatGk =Ck and that the cycle onknodes with each edge of weight λ2 is a k-chordal graph with the desired properties, i.e.,λ(Ck) =λ, Λ(Ck) =k
2
=(k+1)k2 −k, andp(Ck) =k. Also every node x∈Ck defines a minimum cut.
Now we show how to inductively construct the desired family. LetGn = (V, E) be ak-chordal graph onnnodes such that Λ(Gn) =(k+1)n2 −k,p(Gn) =k, andλ(Gn) = λ. Also assume that each nodexinGn defines a minimum cut. We describe how to construct ak-chordal graphGn = (V, E) fromGn, where|V|=n=n+k−2, such that Λ(Gn) = (k+1)n2 −k,λ(Gn) =λ,p(Gn) =k, and every node ofGn defines a minimum cut, thereby proving the existence of the desired family.
Construction of Gn from Gn. Let ube any node in Gn. Then, let V = (V − {u})∪P, where P ={y1, y2, . . . , yk−1} are not already present in V. Let N(u) = {z1, z2, . . . , zl} be the neighbors of u in Gn. Let E = (E − {(u, zi) : 1 ≤ i ≤ l})∪ {(y1, zi) : 1≤i ≤l} ∪ {(yk−1, zi) : 1 ≤i ≤l} ∪ {(yj, yj+1) : 1 ≤j ≤ k−2}, where the weightsw(zi, y1) =w(zi, yk−1) = w(z2i,u) for 1≤i≤l andw(yj, yj+1) = λ2 for 1 ≤ j ≤ k−2. Thus, to get Gn, we remove u from Gn along with the edges incident on it and add a path (y1, y2, . . . , yk−1) with each edge of weight λ2. Also each neighbor zi of u in Gn is now connected to y1 and yk−1. Moreover, the weight of (zi, y1) and (zi, yk−1) will be assigned half the weight of the edge (zi, u) in Gn. It may be noted that the contracted graphs Gn/P =Gn and Gn/P =Ck with each edge having weight λ2.
Claim 9.2. Let (S, S)be a minimum cut ofGn which crosses with the cut(P, P), whereP ={y1, y2, . . . , yk−1}. Then, exactly one of the two edges(zi, y1)or (zi, yk−1) (1≤i≤l) will belong to E(S, S). (Recall that {z1, z2, . . . , zl} are the nodes in Gn
which correspond to the neighbors of uinGn.)
Proof. First, we claim that both the nodesy1 and yk−1 cannot be on the same side of the minimum cut (S, S). Suppose for example, {y1, yk−1} ⊆S. Because all
the edges fromP toP are incident on eithery1oryk−1, E(S∩P, S∩P) =∅. (Note thatS∩P andS∩P will be nonempty since (S, S) is assumed to cross with (P, P).) Therefore the induced subgraph onS will be disconnected, which is a contradiction of Lemma 2.1 since (S, S) is assumed to be a minimum cut. Now without loss of generality assume that y1 ∈ S and yk−1 ∈ S. Then clearly one of the two edges (zi, y1) or (zi, yk−1) (since both these edges exist by construction) will belong to E(S, S).
Claim 9.3. λ(Gn) =λ(Gn) =λand (P, P) is a minimum cut ofGn.
Proof. First note that the cut (P, P) in Gn has weight w(P, P) = λ. This is easily seen from the fact that if we contract P, replacing the set P with the node u, we will get Gn (i.e., Gn/P = Gn), and the cut (P, P) in Gn will correspond to the single vertex minimum cut ({u},{u}) in Gn. Now we will show that every cut in Gn has weight at least λ, thereby establishing that λ(Gn) = λ and (P, P) is a minimum cut of Gn. Suppose λ(Gn) < λ. Then let (S, S) be a minimum cut of Gn. If S (or S) is a subset of P or P, then one of the contracted graphs Gn/P =Gn or Gn/P =Ck will contain a corresponding cut with the same value, which clearly will be a contradiction since λ(Gn) = λ and λ(Ck) = λ. Thus by Lemma 2.3, (S, S) must cross with (P, P) inGn, which means S∩P, S∩P, S∩P, and S∩P are nonempty. Now, by Claim 9.2, exactly one of the two edges (zi, y1) or (zi, yk−1) (1≤i ≤l) will belong toE(S, S). Recall that w(zi, y<
