Classification of interpolation theorems for spanning trees and other families of spanning subgraphs

Classification Interpolation for Spanning and Other Families of Spanning Subgraphs -

of Theorems Trees

Frank Harary DEPARTMENT OF COMPUTER SCIENCE

NEW MEXICO STATE UNIVERSITY U S CRUCES, NEW MEXICO 88003

Michael J. Plantholt DEPARTMENT OF MATHEMATICS

ILLINOIS STATE UNIVERSITY NORMAL, ILLINOIS 67767

ABSTRACT

We say that a graphical invariant i of a graph interpolates over a family 8 of graphs if i satisfies the following property: If rn and M are the minimum and maximum values (respectively) of i over all graphs in 8 then for each k , rn 4 k I M, there is a graph H in 8 for which i (H ) = k . In previous works it was shown that when 8 is the set of spanning trees of a connected graph G, a large number of invariants interpolate (some of these invariants require the additional assumption that G be 2-connected). Although the proofs of all these results use the same basic idea of grad- ually transforming one tree into another via a sequence of edge exchanges, some of these processes require sequences that use more properties of trees than do others. We show that the edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees. This idea is then used to obtain new interpolation results for some invariants, and to show how the exchange 'methods and interpolation results on spanning trees can be extended to other families of spanning subgraphs.

1. INTRODUCTION

A graphical invariant i is said to interpolare over a family 8 of graphs if, for each value k between the minimum and maximum values of i over all graphs in Journal of Graph Theory, Vol. 13, No. 6, 703-712 (1989) 0 1989 by John Wiley & Sons, Inc. CCC 0364-9024/89/060703- 1 0$04.00

704 JOURNAL OF GRAPH THEORY

8, there exists some graph H in 9 such that i ( H ) = k. Several recent results have identified interpolating invariants on the family of spanning trees of a connected graph G. Various subsets of the authors Harary, Lewinter, and Schuster [2-51 have shown that the number of endnodes, the node and edge independence numbers, the domination number, maximum degree, node-edge and edge-node covering numbers, and number of degree-preserving nodes are all interpolating invariants on the family of spanning trees of any connected graph G. The diameter is not an interpolating invariant over all such families: observe that for the graph in Figure 1, each spanning tree has diameter 5 or 7. However, Harary, Mokken, and Plantholt [ I ] have shown that the diameter does interpolate over the set of spanning trees of any graph G that is 2-connected.

An examination of the proofs of these results reveals that they all depend on a sequence of edge exchanges that transform one spanning tree into another but these sequences are not identically constructed, as some rely more heavily on properties of the spanning trees than do others.

We show that the required transformation sequences can be classified as one of three types: simple edge-exchange, adjacent edge-exchange, or leaf edge- exchange (given here in order from weakest to strongest in terms of the properties of the spanning tree families used). This classification observation is then used to list previously unnoted interpolation results on spanning trees for the following invariants: the number of nodes with maximum degree, the path number, and the independent node-edge and node-node domination numbers. Finally, by noting which properties of spanning trees underlie the construction of the three possible types of transition sequences we are able to extend previous interpolation results to new families of graphs, such as all spanning bipartite subgraphs or all connected unicyclic spanning subgraphs of a given graph G.

2. TRANSFORMATION BY EDGE-EXCHANGES

Let G be a connected graph with p labeled nodes, and let 9 denote its family of spanning trees. It is well known that given any two trees To, T* in 9, To can be transformed into T* in at most p - 1 steps as follows. Let e be any edge of T* that is not in To. Then To + e contains a unique cycle with an edgefthat is not in T*; therefore, T, = To + e - f is a spanning tree of G that has one more edge in common with T* than does To. Iterating this procedure, we eventually

G:

FIGURE 1. A graph G for which the diameter does not interpolate over the spanning trees.

CLASSIFICATION OF INTERPOLATION THEOREMS 705

obtain the tree T* from To afterj 5 p - 1 such edge-exchanges. In the process, we have formed a sequence To, Ti, . . . , T, = T* of spanning trees of G. We shall call this sequence a simple edge-exchange (SEE) transition sequence. Such a sequence is depicted in Figure 2 below.

As was noted in [l], it is always possible to obtain a simple edge-exchange transition sequence with the additional property that in each exchange T, + T, + e - f, edges e andf are adjacent in G (although the entire transition sequence may now require more than p - 1 total exchanges); if the sequence has this additional property, we shall call it an adjacent edge-exchange (AEE) transition sequence. To obtain such a sequence, start with any To + T* SEE transition sequence. Then after labeling the unique cycle in T, + e as in Figure 3, replace the simple exchange T, + T, + e - f by the subsequence T,, T, + e - e l , T, + e - e2, . . . , T, + e - f. For the example in Figure 2, the exchange T, + T2 could thus be replaced by Ti, T, + (2,5) - (2,3), T2. It is clear that performing this subsequence substitution for each simple exchange yields the desired AEE transition sequence.

Finally, it was also shown in [ 11 that if G is 2-connected, it is always possible to obtain a To + T* AEE transition sequence that has the additional property that for each edge exchange T, 4 Ti + e - f, edges e andfare incident with a corn- mon leaf node of T, (and thus also of T, + e - f). We shall call such a construction a leaf edge-exchange (LEE) transition sequence.

FIGURE 2. 6, 6 , 5. 5 = T*.

A connected graph G with To + T* SEE transition sequence r=- 0 . 0

e=e, ek= f

0 . 0 0 0 .

FIGURE 3. Labeling of cycle in T, + e.


3. CLASSIFICATION OF INTERPOLATING INVARIANTS ON SPANNING TREES OF A CONNECTED GRAPH

Recall that a SEE transition sequence of spanning trees guarantees only that T,+, = T, + e - f with no additional restrictions on e and f except that T, and T,,, be spanning trees of G. Observe that in order to then prove that an invariant i interpolates over the spanning trees of G, it suffices to show that for any graph H (not necessarily a tree) and edge e in its complement H, either

(1) i ( H ) 5 i(H + e) 5 i ( H ) + 1, or (2) i ( H ) - 1 I i(H + e) 5 i ( H )

The interpolation result then follows by applying (1) or (2) twice: once adding edge e to H = T. and once adding edge f to H = T,+,; together these combine to show that i(TJ - 1 I i(T,+,) 5 i(T,) + 1. We shall call i a positive invariant if it satisfies condition (1) and a negative invariant if it satisfies condition (2). We can then summarize the results in [2] and [3] in the corollary that follows.

Theorem 1. The node independence number Po, node-node domination number am, and edge-node covering number aIo are negative invariants. The edge independence number p, , maximum degree A, and node-edge covering number aol are positive invariants. I

Corollary la. The invariants Po, a,, a,,, PI , A, and aol interpolate over the set of spanning trees of any connected graph. I

Next consider any AEE transition sequence To, TI, . . . , of spanning trees of G. Recall that in each individual edge-exchange --* Ti + e - f, edges e andf are adjacent, say e = uu, f = uw. Thus in each edge exchange, one node degree is increased by one, one node degree is decreased by one, and all other node degrees remain unchanged. We thus obtain the following result for the number of endnodes, which was given in [ 5 ] , and for the number of degree-preserving nodes, which appeared in [4]. The remaining interpolation observations are new. Recall that a node u in G is said to be degree-preserving in a spanning tree T if deg,(u) = deg,(u), and that the path number of a graph is the minimum number of edge-disjoint paths required to cover the edges of the graph.

Theorem 2. The numbers of endnodes, of degree-preserving nodes, of nodes of maximum degree, the edge-edge covering number, and the path number are interpolating invariants over the set of spanning trees of any connected graph G .

Proof. Let To, T,, . . . , T j be any AEE transition sequence of spanning trees of G, and consider one of the edge-exchanges, say T + T, + e - f = T,,,. In the transformation from T to T,+, one node u has its degree increased by one unit, another node w has its degree decreased by one unit, and all other degrees remain unchanged. Since both T, and T,+, have minimum degree one, it is impos-


sible to have either deg&) = dega(w) = 1, or degTt+,(u) = degTi+,(w) = 1. It follows that the numbers of endnodes in Ti and r.+, differ by at most one; therefore, the number of endnodes interpolates over the spanning trees of G. Simi- larly, since the degree of any node in 6 or T,,, is bounded above by its degree in G and also the maximum degree of G, it follows that the number of degree- preserving nodes and the number of nodes with degree A(G) interpolate over the spanning trees. It is likewise easy to check that the edge-edge covering number will not change by more than one in an AEE transition.

T,,, the number of nodes that have odd degree can change by at most two. Using the well-known fact that the path number of a tree equals half the number of nodes with odd degree, we see that the path number also interpolates over the spanning trees of G. I

Finally, it is clear by the previous analysis that in the transition Ti

Let us now turn to the third type of transition sequence: LEE transition sequences. Recall that any 2-connected G contains a LEE transition sequence To, TI, . . . , I;. = T* of spanning trees for any two spanning trees To, T*. There- fore, any invariant whose value changes by at most one under a leaf edge- exchange will interpolate over the spanning trees of any 2-connected graph G. The interpolation result below for the diameter was proved in 111. In [2], it was shown that the node independent domination number does not necessarily interpolate over the spanning trees of an arbitrary connected graph G; a similar result was shown for the node-edge independent covering number.

Theorem 3. If G is a 2-connected graph then the diameter, node independent domination number, and node-edge independent covering number interpolate over the set of spanning trees of G.

Proof. By the previous discussion, it is enough to show that the values of the invariants are changed by at most one by a leaf edge-exchange. Moreover, as the direction of each edge-exchange can be reversed, it suffices to show that a leaf edge-exchange cannot increase the invariant value by more than one. We shall assume that the leaf edge-exchange being considered is that shown in Figure 4, where edge uv is replaced by the edge vw.

FIGURE 4. A leaf edge-exchange T-+ T+,. Node v has degree 1 in both trehs.


Clearly the distance between any two nodes in Ttl equals their distance in T, unless one of the two nodes is u. Moreover, for any node x not equal to u, the distance between u and x in Ttl is one more than the distance between w and x in q. It follows that diam(T,+,) I diam(T,) + 1, so that the diameter interpolates.

To show that the node independent domination number interpolates, let S be any minimum node independent dominating set in T,. To construct a node independent dominating set S' for T,,, with cardinality at most IS I + 1, we consider four cases.

Case 1 . u E S, w e S. If u is adjacent to a node of S other than u, then let S' = S; otherwise, let S' = S U {u}.

Case 2 . u E S , w E S. If u is adjacent to a node of S other than u, let S' = S - {u}; otherwise, let S' = S U {u} - {u}.

Case 3 . u e S, w e S. Here we must have u E S, so let S' = S U {u}.

Case 4 . u S, w E S. Again u E S, so let S' = S.

In each case the independent domination number for T,+I is at most one more than the independent domination number for T, so that the result now follows.

Finally, we note that the verification of interpolation for the node-edge independent covering number is almost identical to that for the node independent domination number. I

4. GENERALIZING INTERPOLATION RESULTS TO OTHER FAMILIES OF SPANNING SUBGRAPHS

It was shown in the previous section that positive and negative invariants interpolate over the spanning trees of any connected graph G . This result holds because between any two spanning trees of G there exists a SEE transition sequence. In fact, this result can be extended in the following form to include other families of spanning subgraphs. We extend the definitions of simple and adjacent edge-exchange transition sequences in the natural way. A simple edge- exchange transition sequence between two graphs F and H over a family 9 of graphs is obtained by a sequence of edge addition and deletion pairs, where the graph obtained after each deletion is' a member of 9; if in each addition/ deletion pair the added edge is adjacent to the edge being deleted, the sequence is an adjacent edge-exchange sequence. The proof below is straightforward.

Theorem 4. Let i be a positive or negative invariant defined on a family % of graphs. Invariant i interpolates over 9 if 9 possesses either of the following two properties:

(1) For any two graphs F and H in 9, there is an SEE transition sequence from F to H consisting entirely of graphs from 9.


(2) For any two graphs F and H in 9, graph H can be obtained from F by a sequence of single edge deletions or additions, with each intermediate graph also in 9. I

Recall that a graphical property P is called hereditary if all subgraphs of a graph with property P must also have property P. Some sample hereditary properties are bipartite, planar, triangle-free, and chromatic number at most k.

We also note that the connectivity and edge-connectivity of a graph are positive invariants. These invariants are not interesting on trees, where their values are always 0 or 1. However, they do become interesting on families of graph that may have higher connectivity.

Corollary 4a. The connectivity, edge-connectivity, node independence number, domination number, edge covering number, edge independence number, maximum degree, and node-edge covering numbers interpolate over the following families of subgraphs of any graph G:

(i) The set of all spanning subgraphs that have a given number k of edges. (ii) The set of all spanning subgraphs that have at least q, and at most q2

edges, where q, and q2 are given integers. (iii) The set of all spanning subgraphs that have maximum degree at most

M, or the set of all spanning subgraphs that have minimum degree at least m, for given integers M and m.

(iv) The set of all spanning subgraphs having a given hereditary property P. (v) The set of all connected spanning k-edge subgraphs of G.

Proof. It is easy to see that the families in (i) satisfy condition (1) of Theo- rem 4, and the families in (ii) and (iii) satisfy condition (2). To verify that the families in (iv) also satisfy condition (2) of Theorem 4, we simply note that the required transition sequence from F to H can be obtained by deleting the edges of F one at a time until we have deleted all edges, and then inserting the edges of H one by one. Since P is hereditary, all graphs obtained during the sequence will also have property P.

Finally, we show that the family 9 of (u) satisfies condition (1) of Theorem 4. Suppose we are looking for a simple edge-exchange transition sequence from graph F to graph H. Let To and T* be spanning trees of F and H, respectively, that have the maximum possible number of edges in common. Let So denote the set of all edges of F that are not in To, and similarly, let S* denote the set of all edges of H - E(T*).

Note that So U S* contains no edges from either To or T*. For suppose that an edge e is in both To and S*. Then there is an edgefof T* such that T* + e - f is a spanning tree of H that includes one more edge of To than does T*, which is impossible. A like argument verifies that no edge is in both T* and So.

Now construct the transition sequence as follows. Successively remove from F each edge in So - S*, and replace each with a different edge in S* - So.


Each of the graphs obtained in this sequence is connected since each contains To. Then complete the transition from F to H by constructing a To + T* SEE transition sequence (carrying along, of course, all edges of S*) that uses the minimum possible number of exchanges. This gives the desired F + H transition sequence through members of 9. I

Finding generalized families of graphs that always possess AEE transition sequences is a more difficult problem. At first glance this property on the family of spanning trees seems to depend only on connectedness, so that it is natural to suspect that the family of spanning connected k-edge subgraphs of G would possess the necessary AEE transition sequences. That this is not the case is illustrated in Figure 5 , which gives a graph for which the family of connected spanning subgraphs with seven edges does not have this property.

The reason that this property on spanning trees fails to generalize to arbitrary connected spanning k-edge subgraphs is that each edge-exchange for spanning trees will necessarily exchange edges that share a common cycle; this property keeps all graphs in the transition sequence connected, a property that breaks down for the unicyclic family of the graph in Figure 5. However, if we add the condition that G is 2-edge-connected, that is, G is connected and has no bridges (cutedges), this difficulty can be overcome.

Theorem 5. Let G be a 2-edge-connected graph, and let % be the family of all connected k-edge spanning subgraphs of G. For any two graphs F,H in 8, there is a transition sequence over 9 of adjacent edge-exchanges from F to H.

Proof. By the proof of part (v) of Corollary 4a, there is a sequence of edge- exchanges transforming F to H via members of 8. Thus it suffices to show that a single such edge-exchange F F + e - f can be achieved by a subsequence of adjacent edge-exchanges through members of 8.

We proceed by induction on the number of interior edges in a shortest path in F + e which begins and ends with edges e and f. We shall call this number the distance between e andfin F + e, denoted disFtc(e,f), and denote the edges of the corresponding shortest path P by e = e,, e , , . . . , ej = f.

If dis,+,(e,f) = 0 then the edges e and f are adjacent, so that F 4 F + e - f is itself the desired edge exchange.

Now assume that F + F + e - f can be written as a subsequence of adjacent edge exchanges through members of 9 whenever dis,,,(e,f) I s, and consider

G:

FIGURE 5. A graph G for which there is no AEE transition sequence from G - e, to G - q through the connected spanning unicyclic subgraphs.

CLASSIFICATION OF INTERPOLATION THEOREMS 71 1

a valid simple edge-exchange F --* F + e - f in which disF+c(e,f) = s + 1. We consider two cases.

Case 1. F + e - el is connected. In this case we begin the subsequence with F, F + e - el, and need to complete it with a subsequence for F + e - el -+ F + el - f. But disF+,(e,,f) = s, so the desired subsequence is guaranteed by the induction hypothesis.

Case 2. F + e - el is disconnected. In this case edge el is a cutedge in F. Since graph G is 2-edge-connected, there is an edge e* of G that is incident with one node in each of the two components of F - e l . Then F + e* has a cycle C* that contains both el and e* (see Figure 6); let the edges of C* be

fl = el ,&. . . ,A-.l,f, = e*, . . . ,& wheref, is adjacent to e, and e, and& is adjacent to el but not adjacent to e.

Since F + e - f is connected and el is a cutedge of F + e, the cycle C* may be chosen so that it does not contain edgef. Let r be the smallest subscript greater than one such that f, is adjacent to an edge of P other than e or e,; clearly r 2 i. We construct the desired F + F + e - f sequence by combining the following three subsequences.

Subsequence 1. Use the AEE sequence F, F + fi - fi+l, F + - f,+*, . . . , F + f, -f, to achieve the transition F + F + f, - f,. Each graph in the sequence is connected since each is obtained by removing an edge in the cycle C* from the connected graph F + e*.

Subsequence 2. It is easy to check that F + f, - f is connected, because F and F + e - f are connected, and el is a cutedge whose removal from F + e leaves e and f in different components. Therefore by the induction hypothesis there is an AEE transition sequence F + fi - f, + F + f, - f since dis,+,(f,,f) 5 s.

e 7- eJ = f - - J-1 . . . . f, .I.. . , --i'

f3

FIGURE 6. Writing F-, F + e - f via an AEE transition sequence. Edges e and e* are in G, but not in F.


Subsequence 3. Use the AEE sequence F + fi - f, F + J - f + e - f2, F + fi - f + e - f3,. . . , F + fi - f + e - f , = F + e - f to achieve the transition F + J - f -+ F + e - f . Again each graph appearing internally in this sequence is connected since each is obtained from the connected graph F +

Finally, the desired AEE transition sequence is obtained by concatenating the three subsequences. I

- f + e by removing an edge of cycle C*.

As in Theorem 2, we immediately obtain the following:

Corollary 5a. The numbers of endnodes, of degree-preserving nodes, of nodes with degree A(G), the edge-edge covering number, and the path number are interpolating invariants over the family of connected k-edge spanning subgraphs of any graph G for any integer k. I

We conclude by noting that we know of no generalization of the LEE transition sequence result on spanning trees to other natural families of spanning subgraphs.

References

[ I ] F. Harary, R. J. Mokken, and M. Plantholt, Interpolation theorem for diameters of spanning trees. IEEE Trans. Circuits Sysr. CAS-30 (1983)

[2] F. Harary and S. Schuster, Interpolation theorems for the independence and domination numbers of spanning trees. Graph Theory in Memory of G . A . Dirac, to appear.

[3] F. Harary and S. Schuster, Interpolation theorems for invariants of spanning trees of a given graph: Covering numbers. The 250rh Anniversary Confer- ence on Graph Theory, Congressus Numerantium, Utilitas Mathematica.

[4] M. Lewinter, Interpolation theorem for the number of degree-preserving vertices of spanning trees. ZEEE Truns. Circuits Syst. CAS-34 (1987) 205.

[ 5 ] S. Schuster, Interpolation theorem for the number of end-vertices of spanning trees. J . Graph Theory 7 (1983) 203-208.

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Classification of interpolation theorems for spanning trees and other families of spanning subgraphs