Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies

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1 Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies Florent R. Madelaine and Iain A. Stewart Department of Computer Science, Durham University, Science Labs, South Road, Durham DH LE, U.K. Abstract We improve upon the best known upper and lower bounds on the sizes of minimal feedback vertex sets in butterflies. Also, we construct new feedback vertex sets in grids so that for a large number of pairs (n, m), the size of our feedback vertex set in the grid M n,m matches the best known lower bound, and for all other pairs, apart from when n or m is exactly 5 and the other parameter is at least 5, it differs from this lower bound by at most. Keywords: feedback vertex sets; grids; butterflies. Introduction A feedback vertex set in an undirected graph is a subset of vertices the removal of which (along with their incident edges) results in an acyclic graph. The feedback vertex set problem is to find a feedback vertex set of minimum cardinality in a graph G, with the size of such a set known as the feedback vertex number τ(g). Whilst the feedback vertex set problem is NP-hard in general, it has been extensively studied in a wide variety of restricted classes of graphs and shown to be polynomial-time solvable in many of these classes. Furthermore, a number of lower and upper bounds on the feedback vertex number of graphs from these classes have been established. The reader is referred to [] for an extensive survey of feedback vertex set problems which ranges over polynomial solvable cases, approximation algorithms, exact algorithms, practical heuristics and applications. In this paper, we are concerned with the classes of graphs known as grids and butterflies. Such graphs are common in the study of interconnection networks for parallel processing as they have particularly attractive properties in this regard (see, for example, []). The study of feedback vertex sets in grids and butterflies has traditionally gone hand-in-hand. In [5], Luccio proved upper and lower bounds on the sizes of minimal feedback vertex sets in both grids and butterflies. It was shown in [5] that the feedback vertex number of the grid M n,m is at most and at least mn + m + n + o(m,n) (m )(n ) +,

2 and that the feedback vertex number of the butterfly B d is at most (see the analysis in []) and at least (d + )d + d d + (definitions of M n,m and B d follow). Subsequently, in [], Caragiannis, Kaklamanis and Kanellopoulos improved the state of affairs by establishing a general lower board technique by which they showed that the feedback vertex number of the butterfly B d is at least (d ) d +. They also showed that the feedback vertex number of the grid M n,m is at most mn m + n 5 and that the feedback vertex number of the butterfly B d is at most (d + )d. Finally, more recently, Chang, Lin and Lee [] both improved Luccio s analysis of the sizes of feedback vertex sets in butterflies and exhibited an algorithm which constructed a feedback vertex set in B d of size (d )d +, if d is even, and of size (d )d + d + d, if d is odd. In this paper, we improve upon Chang, Lin and Lee s algorithm and obtain a smaller upper bound on the feedback vertex number of a butterfly B d, when d 5. Our algorithm is very similar to that of Chang, Lin and Lee except that our starting point in the recursive algorithm is improved feedback vertex sets for B 5 and B. We find that we can use Chang, Lin and Lee s analysis to prove our algorithm correct and also to establish our improved bounds. We also improve upon Luccio s lower bound on the feedback vertex numbers of butterflies. As regards grids, we make dramatic progress. We construct new feedback vertex sets in grids so that for a large number of pairs (n,m), the size of our feedback vertex set in the grid M n,m matches the best known lower bound (from [5]), and for all other pairs, apart from when n or m is exactly 5 and the other parameter is at least 5, it differs from this lower bound by at most. This paper is structured as follows. In Section, we provide the basic definitions, before dealing with feedback vertex sets in grids in Section and in butterflies in Section. Our conclusions are given in Section 5.

3 Basic definitions Let n,m. The rectangular grid with n rows and m columns (or n m mesh), denoted by M n,m, is the graph with vertex set V (M n,m ) defined as {v i,j : 0 i < n,0 j < m} and edge set E(M n,m ) defined as {(v i,j,v i+,j ) : 0 i < n,0 j < m} {(v i,j,v i,j+ ) : 0 i < n,0 j < m }. Let d. The d-dimensional butterfly B d has vertex set V (B d ) partitioned into (d + ) rows, whereupon each row contains d vertices. Every vertex of V (B d ) is indexed by the pair (i,j) where i indicates its row and j its column in that row: as such, we refer to the vertices of V (B d ) as {v i,j : 0 i d,0 j d }. The edge set E(B d ) of B d consists of the following edges. For every pair of adjacent rows, there is an edge joining corresponding vertices, v i,j and v i+,j, on these two rows; that is, there are edges {(v i,j,v i+,j ) : 0 i < d,0 j d }. For every pair of adjacent rows, there is an edge joining a vertex v i,j on the lower-indexed row to the vertex v i+,ji on the higher-indexed row so that the binary representation of the integer j i differs from that of the integer j only in the ith position (where the right-most bit is bit 0); that is, there are edges {(v i,j,v i+,ji ) : 0 i < d,0 j d }. Grids and butterflies can be visualized as in Figs. and 8, respectively. We adopt the following notation (in line with that of Chang, Lin and Lee in []). If G is a graph with vertex set V (G) and V is a subset of vertices of V (G) then the subgraph of G induced by the vertices of V is denoted G[V ] and the subgraph of G induced by the vertices of V (G) \ V is denoted G \ V. New upper bounds for grids In this section, we derive new upper bounds on the sizes of minimal feedback vertex sets in two-dimensional grids. For a large number of pairs (n,m), our upper bound on the size of a minimal feedback vertex set of M n,m matches the lower bound from [5], and on the other pairs, apart from when n or m is exactly 5 and the other parameter is at least 5, it differs from this lower bound by at most. Case (i) Let n be such that n (mod ), and let m be even. Define the set of vertices X n,m of V (M n,m ) as A n,m B n,m C n,m D n,m E n,m F n,m where: A n,m = {v i,j : 0 i n,i (mod ), j m,j even}; B n,m = {v i,j : 0 i n,i (mod ), j m,j odd}; C n,m = {v i,j : 0 i n,i (mod ), j m,j odd};

4 D n,m = {v i,j : 0 i n,i 5 (mod ), j m,j even}; E n,m = {v i, : 0 i n,i 0 (mod )}; F n,m = {v i,m : 0 i n,i (mod )}. The set X 0,8 is shown in Fig.. 0 row column Figure. The set of vertices X 0,8. We claim that X n,m is a feedback vertex set. Observe that if there is a cycle in M n,m \X n,m then the inclusion of the vertices of A n,m B n,m C n,m D n,m in X n,m means that the cycle must use only the perimeter vertices of M n,m or the vertices on row i, for each i 0 (mod ). However, the vertices of E n,m and F n,m preclude any such cycle, and so X n,m is a feedback vertex set of M n,m. The size of X n,m is while Luccio s lower bound is (n ) (n ) (n )(m ) (m ) + + = +, (m )(n ) + which, for n (mod ) and m even, is identical to the size of X n,m. Hence, X n,m is a feedback vertex set of minimal size. Before continuing, let us look at the feedback vertex set X n,m and why it is of minimal size from a different perspective (this perspective underpins Luccio s lower bound construction in [5] but will be of use to us later in alternative contexts). Consider the perimeter-cycle of M n,m ; this cycle must contain at least one vertex from any feedback vertex set, so choose such a vertex to be v 0,. After removing v 0, and any incident edges from M n,m, to get M n,m, there is a natural perimeter-cycle which is as before except that the sub-path navigating around v 0, is,...,v,0,v,0,v,,v,,v 0,,v 0,,...

5 Also note that the sub-graph of the original M n,m induced by the vertices strictly outside the perimeter-cycle but not vertices of our partial feedback vertex set (at present, just the vertex v 0,0 ) is acyclic. Similarly, the perimeter-cycle of M n,m contains at least one vertex of any resulting feedback vertex set; so choose the vertex v, to be such a vertex. In M n,m, obtained from M n,m by removing v, and any incident edges, there is a natural perimeter cycle and the sub-graph of the original M n,m induced by the vertices strictly outside the perimeter-cycle but not vertices of our partial feedback vertex set (at present, just the vertices v 0,0 and v 0, ) is acyclic. Similarly, the perimeter-cycle of M n,m contains at least one vertex of any resulting feedback vertex set; so choose the vertex v, to be such a vertex. In M n,m, obtained from M n,m by removing v, and any incident edges, there is a natural perimeter cycle and the sub-graph of the original M n,m induced by the vertices strictly outside the perimeter-cycle but not vertices of our partial feedback vertex set (at present, just the vertices v 0,0 and v 0, ) is acyclic. Continuing in this fashion ultimately results in a perimeter-cycle the breaking of which results in an empty perimeter-cycle, so that the sub-graph of M n,m induced by those vertices not in the resulting feedback vertex set is indeed acyclic. This constructive approach to the formation of X n,m can be visualized as in Fig., where the order in which perimeter-cycle vertices are chosen is given and where at each stage, the edges incident with the chosen vertex and outside the resulting perimeter-cycle are omitted. Yet another alternative way of viewing the construction of X n,m is via a tessellation of the grid, in a natural way. Note that given any feedback vertex set X, a feedback vertex set Y X can be constructed by adopting the above procedure and making appropriate choices Figure. The set of vertices X 0,8 formed by perimeter-breaking. Apart from the first and last choices of the vertices of X n,m in the above procedure, the choices are optimal in the sense that at any stage, no other choice could decrease the number of cells inside the perimeter-cycle more than the number resulting from the vertex chosen (the most the number of cells can decrease by is, as is the case with our choices). Indeed, the first choice of vertex is optimal in this sense too (with 0 5

6 a decrease of cells). Initially, there are (n )(m ) cells, and any first choice decreases this number to at least (n )(m ). Subsequent choices decrease this number by at most cells per choice, and so as n (mod ), after (n )(m ) choices, there is at least cell inside the perimeter-cycle. Thus, the size of any feedback vertex set is at least (n )(m ) +. Case (ii) Let n be such that n (mod ), and let m 7 be odd. Partition M n,m into two sub-grids, one, call it M n,m, induced by the vertices in columns 0, and, and one, call it M n,m, induced by the vertices in the remaining columns. Note that M n,m is such that it has an even number of columns. There are two cases, depending upon whether n (mod ) or not. Sub-case (ii.a) n (mod ). We can build a set of vertices X n,m in M n,m, as above, except starting from the right-hand side as opposed to the left. In particular, define X n,m as A n,m B n,m C n,m D n,m E n,m F n,m where: A n,m = {v i,j : 0 i n,i (mod ), j m,j even}; B n,m = {v i,j : 0 i n,i (mod ),5 j m,j odd}; C n,m = {v i,j : 0 i n,i (mod ),5 j m,j odd}; D n,m = {v i,j : 0 i n,i 5 (mod ), j m,j even}; E n,m = {v i,m : 0 i n,i 0 (mod )}; F n,m = {v i, : 0 i < n,i (mod )} {v n, }. In M n,m, define X n,m as A n,m B n,m {v 0, } where: A n,m = {v i, : i n,i even}; B n,m = {v i, : i n,i odd}. Define X n,m = X n,m X n,m. The construction of X 0, by perimeter-breaking can be visualized as in Fig.. The perimeter-breaking argument applied above yields that X n,m is a feedback vertex set of M n,m. Case (ii.b) n (mod ). We can build a set of vertices X n,m in M n,m, as above (starting from the left-hand side). In particular, define X n,m as A n,m B n,m C n,m D n,m E n,m F n,m where:

7 Figure. The set of vertices X 0, formed by perimeter-breaking. A n,m = {v i,j : 0 i n,i (mod ),5 j m,j odd}; B n,m = {v i,j : 0 i n,i (mod ), j m,j even}; C n,m = {v i,j : 0 i n,i (mod ), j m,j even}; D n,m = {v i,j : 0 i n,i 5 (mod ),5 j m,j odd}; E n,m = {v i, : 0 i < n,i 0 (mod )}; F n,m = {v i,m : 0 i n,i (mod )} {v n, }. In M n,m, define X n,m as A n,m B n,m {v 0, } where: A n,m = {v i, : i n,i odd}; B n,m = {v i, : i n,i even}. Define X n,m = X n,m X n,m. The construction of X, by perimeter-breaking can be visualized as in Fig.. Again, the perimeter-breaking argument applied above yields that X n,m is a feedback vertex set of M n,m. In both of the above cases, the size of X n,m is (n )(m ) + + (n ) = (n )(m ) and so X n,m is a minimal feedback vertex set as if we denote the lower board on the size of a minimal feedback vertex set of M n,m obtained by Luccio in [5] as lb n,m then in this case X n,m = lb n,m. Case (iii) Let n be such that n 0 (mod ) and let m be even such that m (mod ) (for if m (mod ) then we can apply either Case (i) or Case (ii)). Let M n,m be the sub-grid induced by the vertices in rows 0,,...,n. Using the construction from Case (i) in M n,m, starting (in the sense of the perimeter-breaking +, 7

8 exposition) with either v 0, or v 0,m, as appropriate, we can build a feedback vertex set X n,m of M n,m of size (n )(m ) + so that v n,m lies in this feedback vertex set Figure. The set of vertices X, formed by perimeter-breaking. Let m be such that m (mod ) and either m = m + or m = m +. Let M n,m be the sub-grid induced by the vertices in rows n,n 5,...,n and in columns 0,,...,m. Using the construction from Case (i) in M n,m, starting with either v n 5,0 or v n,0, as appropriate, we can build a feedback vertex set X n,m of M n,m of size 5(m ) + so that v n,m lies in this feedback vertex set. Consider the partial feedback set X n,m X n,m (note that X n,m X n,m is a feedback vertex set of the sub-grid of M n,m induced by the vertices of M n,m and M n,m). If m = m (with m 0 (mod )) then the additional vertices v n,m, v n,m and v n,m extend this set to a feedback vertex set of M n,m. If m = m (with m (mod )) then the additional vertex v n,m extends this set to a feedback vertex set of M n,m. The constructions are illustrated for M, and M,8 in Fig. 5. Consequently, we have constructed a feedback vertex set of size and of size (nm n m + ) (nm n m + 5) = lb n,m + if m 0 (mod ) = lb n,m + if m (mod ). 8

9 m' = m - m' = m - Figure 5. The sets of vertices X, and X,8. Case (iv) Suppose now that n is such that n 0 (mod ) and m is odd such that m (mod ). Let M n,m be the sub-grid induced by the vertices in rows 0,,...,n. Using the constructions in Case (ii), we can build a feedback vertex set X n,m of M n,m of size (n )(m ) so that either v n,m or v n,m lies in this feedback vertex set. Defining M n,m as we did in Case (iii), we can build a feedback vertex set X n,m of M n,m of size 5(m ) so that v n,m lies in this feedback vertex set. Consider the partial feedback set X n,m X n,m (note that X n,m X n,m is a feedback vertex set of the sub-grid of M n,m induced by the vertices of M n,m and M n,m). If m = m then the additional vertices v n 5,m, v n,m and v n,m extend this set to a feedback vertex set of M n,m. If m = m then the additional vertices v n 5,m and v n,m extend this set to a feedback vertex set of M n,m. The constructions are illustrated for M, and M, in Fig.. Consequently, we have constructed a feedback vertex set of size + + and of size (nm n m + ) (nm n m + 8) = lb n,m + if m 0 (mod ) = lb n,m + if m (mod ).

10 m' = m - m' = m - Figure. The sets of vertices X, and X,. Case (v) Let n be such that n (mod ) and let m be even such that m (mod ). Let M n,m be the sub-grid induced by the vertices in rows 0,,...,n 8. Using the construction in Case (i), we can build a feedback vertex set X n,m of M n,m of size (n 8)(m ) so that v n 8,m lies in this feedback vertex set. Let m be such that m (mod ) and either m = m + or m = m +. Let M n,m be the sub-grid induced by the vertices in rows n 8,n 7,...,n and in columns 0,,...,m. Using the construction from Case (i) in M n,m, we can build a feedback vertex set X n,m of M n,m of size 7(m ) so that v n,m lies in this feedback vertex set. Consider the partial feedback set X n,m X n,m (note that X n,m X n,m is a feedback vertex set of the sub-grid of M n,m induced by the vertices of M n,m and M n,m). If m = m then the additional vertices v n,m, v n 5,m, v n,m and v n,m extend this set to a feedback vertex set of M n,m. If m = m then the additional vertices v n,m and v n,m extend this set to a feedback vertex set of M n,m (the situation can be visualized using similar figures to those already detailed, hence we omit them). Consequently, we have constructed a feedback vertex set of size and of size (nm n m + 5) (nm n m + ) + + = lb n,m + if m 0 (mod ) = lb n,m + if m (mod ). 0

11 Case (vi) Let n be such that n (mod ) and let m 7 be odd such that m (mod ). Let M n,m be the sub-grid induced by the vertices in rows 0,,...,n 8. Using the constructions in Case (ii), we can build a feedback vertex set X n,m of M n,m of size (n 8)(m ) so that either v n 8,m or v n 8,m lies in this feedback vertex set. Let m be such that m (mod ) and either m = m + or m = m +. Let M n,m be the sub-grid induced by the vertices in rows n 8,n 7,...,n and in columns 0,,...,m. Using the construction from Case (i) in M n,m, we can build a feedback vertex set X n,m of M n,m of size 7(m ) so that v n,m lies in this feedback vertex set. Consider the partial feedback set X n,m X n,m (note that X n,m X n,m is a feedback vertex set of the sub-grid of M n,m induced by the vertices of M n,m and M n,m). If m = m then the additional 5 vertices v n 7,m, v n 5,m, v n,m, v n,m and v n,m extend this set to a feedback vertex set of M n,m. If m = m then the additional vertices v n 7,m, v n 5,m and v n,m extend this set to a feedback vertex set of M n,m. Consequently, we have constructed a feedback vertex set of size (nm n m + 8) + + = lb n,m + if m 0 (mod ) and of size (nm n m + ) = lb n,m + if m (mod ). Drawing together the results of this section, we obtain the following theorem. Theorem If the pair (n,m) does not lie in the set {(,m),(n,) : n,m } {(,m),(n,) : n,m } {(,5),(5,)} {(5,m),(n,5) : n,m 5} {(,),(,8),(8,),(8,8)} then the size of a minimal feedback vertex set in the grid M n,m is lb n,m, lb n,m + or lb n,m +. Given any specific pair (n,m) for which Theorem is relevant, an upper bound on the size of the minimal feedback vertex set can be read from the appropriate case considered earlier. Ignoring the finite number of isolated grids for which Theorem does not apply (for in each of these cases the dimensions are sufficiently small for a simple computer program to find the size of a minimal feedback vertex set), we are left with three (infinite) classes of grids lying outside our analysis. However, for the first of these

12 classes we can resolve the situation exactly: when (n,) {(n,) : n }, the size of a minimal feedback vertex set is, trivially, n. We shall return to the other two classes presently, after we have examined an alternative feedback vertex set construction. From the constructions above, we have yet to exhibit minimal feedback vertex sets for certain grid dimensions, i.e., when neither n nor m is equivalent to modulo. However, we have another construction which enables us to construct a minimal feedback vertex set in some of these cases. Moreover, our construction also allows us to use feedback vertex sets in smaller grids to build feedback vertex sets in larger grids where the size of the constructed feedback vertex set is controlled in terms of the size of the original feedback vertex set. We can expand the grid M n,m by: placing a new edge-vertex in the middle of each edge of M n,m ; placing a new cell-vertex in the middle of each cell of M n,m ; and joining each new cell-vertex to the new edge-vertices on the perimeter of its cell. Note that the expanded grid, which we denote E(M n,m ), is actually a copy of M n,m. Let X be a feedback vertex set of M n,m. We expand M n,m into E(M n,m ) and define the set of vertices E(X) to consist of the vertices corresponding to the vertices of X in union with the set of cell-vertices of E(M n,m ). It is immediate that the set E(X) is a feedback vertex set of E(M n,m ) (essentially, if we remove the cell-vertices from E(M n,m ) then cycles correspond to cycles in M n,m, and vice versa). The construction can be visualized as in Fig. 7, where the white vertices in M 0,0, on the right, are vertices of its feedback vertex set corresponding to the vertices of the feedback vertex set in M 5,5, on the left, and the grey vertices in M 0,0 are the added cell-vertices Figure 7. Expanding a grid with a feedback vertex set. The size of the feedback vertex set E(X) of E(M n,m ) is equal to the size of the feedback vertex set X of M n,m plus (n )(m ). That is, E(X) = X + (n )(m ).

13 Luccio s lower bounds lb n,m and lb n,m on the sizes of minimal feedback vertex sets of M n,m and M n,m are nm n m + nm n m + 5 and, respectively. Hence, E(X) lb n,m = X lb n,m. Thus, the distance a feedback vertex set is away from the lower bound lb n,m in M n,m is preserved by the construction in M n,m. In particular, if X is a minimal feedback vertex set of M n,m of size lb n,m then E(X) is a minimal feedback vertex set of M n,m, of size lb n,m. The feedback vertex set of M 5,5 shown in Fig. 7 is minimal (and has size lb 5,5 ), thus we have effectively constructed minimal feedback vertex sets in all grids M r +, r +, for r. Our construction generalizes, yet simplifies, the construction of Luccio in [5]. Let us now return to M,n, where n. As M,n = E(M,n ), we immediately obtain an upper bound of (n ) for τ(m,n ). Consider the grid M,5 and how many of the vertices in columns 0,, and must necessarily lie in a minimal feedback vertex set of M,5 : a simple case-by-case analysis yields that at least such vertices must do so. Divide M,n, where n, into copies of M,5, the first copy consisting of the vertices in columns 0,,, and, the second copy of vertices in columns, 5,, 7 and 8, the third copy of vertices in columns 8,, 0,, and, and so on. By above, at least of the vertices of any feedback vertex set of M,n must lie in columns 0,, and, at least must lie in columns, 5, and 7, at least must lie in columns 8,, 0 and, and so on. Hence, if n is odd then and if n is even then τ(m,n ) τ(m,n ) (n ) (n ) = (n ), (n ) + = (in the latter case, we divide M,n into copies of M,5 and we need at least vertices to break cycles involving vertices in the left-over columns indexed by n, n and n ). Thus, when n, (n ) τ(m,n ) =. Trivially, when n, (n ) (n ) τ(m,n ) +

14 (simply consider the copy of M,n induced by the vertices of M,n in columns 0,,...,n ). Finally, we are left with the grids {M 5,n : n 5}. Here, unfortunately, the situation is less clear. Consider the grid M 5,n where n and where n is odd. Divide M 5,n into copies of M 5,5, just as we did with M,n and M,5 above. In each copy of M 5,5, include the vertices (isomorphic to) v,, v,, v,, v, and v, in a partial feedback set. After removing these vertices and their incident edges from M 5,n, what remains is to, all intents and purposes, essentially a copy of M, n +. Thus, τ(m 5,n ) is at most 5(n ) n (n ) + = 8 When n is even, we proceed similarly to the above except that there are the vertices in the columns indexed by n, n and n to consider too. It is not difficult to see that that additional vertices suffice to obtain our feedback vertex set. Hence, τ(m 5,n ) is at most 5(n ) n =. n 0 (n ) = (as n is even). Consider the grid M 5,n, where n. There are two cases: either divides n (that is, n is odd) or divides n (that is, n is even). Considering M 5,n as either M 5,n, with an extra column of vertices, or as M 5,n, with an extra columns of vertices, results in a feedback vertex set of size (n ) and of size (n ) + = + 5 = n n That is, when n, τ(m 5,n ) is at most n 0., if n is odd,, if n is even. Whilst our upper bound on the size of a minimal feedback vertex set of the grid M 5,n, where n 5, improves upon the best known upper bound from [], we have been unable to improve upon the best known lower bound from [5], namely lb 5,n = 8n 7 or lb 5,n = 8n. Consequently, we can draw all our results regarding grids together in the following theorem. Theorem There exists a computable function f(n,m) such that the size of a minimal feedback vertex set of the grid M n,m, where (n,m) {(n,m) : n,m } \ {(n,m) : (n = 5 and m 5) or (m = 5 and n 5)}, is one of f(n,m), f(n,m) + or f(n,m) +.

15 Of course, the word computable, whilst strictly correct, is somewhat inappropriate as the function f can be described very concisely according to the different cases arising in this section. Also, τ(m n,m ) is known exactly for a lot of different cases. Feedback vertex sets in butterflies In this section, we improve the known upper and lower bounds on the size of minimal feedback vertex sets in butterflies. We begin with some basic structural decompositions. If 0 j d then denote by bit(d,j) the bit-string of length d that is the binary representation of j. Also, for any bit-string b of length d, denote by bin(b) the integer whose binary representation as a bit string of length d is b. For any two bit-strings b and b, denote the concatenation of b and b as bb. Fix some d. Let b be a bit-string whose length, b, is at least and at most d. Define the subgraph B b d of B d as the subgraph of B d induced by the vertices of the set {v i,j : b i d,0 j d,j bin(b) mod b }; that is, the vertices on rows b, b +,...,d whose column names (when written in binary) end in b. The sub-graphs B 0 (with edges in bold) and B 0 (with dashed edges) are illustrated in Fig Figure 8. Butterflies contained within butterflies. Lemma Let d and let b be a bit-string whose length is at least and at most d. The subgraph B b d of B d is isomorphic to B d b via the isomorphism β b : B d b B b d given by β b(v i,j ) = v i,j, with i = i + b and j = bit(d b,j)b. Proof For Bd 0 and B d (that is, for the bit-strings b = 0 and b = ), the definition of B d yields the result. For other bit-strings, the result then follows by a simple induction. Lemma Let d and let the set of vertices U of B d be defined as {v i,j : i = 0,,...,d,0 j d }. The subgraph B d [U] of B d consists of two disjoint copies 5

16 B l d and Br d of B d where the isomorphisms β l : B d B l d and β r : B d B r d are given by β l (v i,j ) = v i,j and β r (v i,j ) = v i,j+ d. Proof Immediate from the definition of B d. We are now in a position to improve the lower bound on the size of a minimal feedback vertex set of B d as established in [], namely lb CKK d = (d )d +. Our improvement is obtained by refining the general theorem used in [] to obtain this lower bound. Proposition 5 Let G = (V,E) be a graph of maximal degree d and let F be a feedback vertex set of G, with H the subgraph G\F. Let F i be the set of vertices in F of degree i in G, let P be the set of edges of G induced by the vertices of F and let c be the number of connected components of H. Then E (Σ d i=i F i ) + P = V F c. Proof The number of edges of E incident with a vertex of F is (Σ d i=i F i ) P. As F is a feedback vertex set, H is a forest and so the number of edges of H is equal to the number of vertices of H minus c; that is, and the result follows. E (Σ d i=i F i ) + P = V F c By applying Proposition 5 to B d, we obtain that (with the definitions as in the statement of Proposition 5), F F = (d ) d + P + c. Note that as P and F are at least 0 and c is at least, we obtain the lower bound lb CKK (d) on the size of a feedback vertex set of B d, as was done in []. However, we can use Proposition 5 to obtain an improved lower bound on the size of a feedback vertex set of B d. Proposition Let d. Any feedback vertex set of B d has size at least (d ) d +. Proof Consider B d, for d, and let F be a minimal feedback vertex set of B d. Let H be the subgraph B d \ F of B d. We may assume that there are no vertices of F on row 0 nor on row d (as if, for example, the vertex v 0,0 is in F then we can replace v 0,0 in F with the vertex v,0 and still obtain a minimal feedback vertex set).

17 By Lemma, the subgraphs Bd l and Br d of B d are isomorphic to B d. Also, as every cycle (of length ) in B d involving only vertices on rows d and d must contain at least one vertex of F (with such a vertex of F being on row d ), there is no path in B d \ F from a vertex of Bd l \ F to a vertex of Br d \ F (also, note that Bl d \ F and Bd r \ F are both non-empty, as no vertex on row 0 is in F). Consider Bd l. By Lemma, Bl d B0 d and Bl d B d are isomorphic to B d. Moreover, as every cycle (of length ) in B d involving only vertices on rows 0 and must contain at least one vertex of F (with such a vertex of F being on row ), there is no path in B d \ F from a vertex of Bd l B0 d to a vertex of Bl d B d. Furthermore, as d, both (Bd l B0 d ) \ F and (Bl d B d ) \ F are non-empty. Similar reasoning applies to (Bd r B0 d ) \ F and (Br d B d ) \ F. Thus, B d \ F consists of at least connected components. Putting c into the equation in Lemma 5 (with G taken as B d ) yields the result. So, if we denote our new lower bound from Proposition as lb d then we have that lb d = lb CKK d +. Whilst our lower bound improvement is somewhat slight, we can make a more significant improvement on the best known upper bound on the size of a minimal feedback vertex set of B d. Definition 7 Define V l = V (B l d) = {v i,j : 0 i d,0 j d }, and for all d. V r = V (B r d) = {v i,j : 0 i d, d j d } V d = {v d,j : 0 j d }, Note that V (B d ) = V d V l V r. Definition 8 Define V d = {v d,j : 0 j d }, and for all d >. V d = {v d,j : d j d }, V d = {v d,j : d j. d } V d = {v d,j :. d j d }, Definition Define V l,0 = {v i,j : 0 i d,0 j d }, V l, = {v i,j : 0 i d, d j d }, 7

18 and for all d >. V r,0 = {v i,j : 0 i d, d j. d } V r, = {v i,j : 0 i d,. d j d }, Note that B d [V l,0 ], B d [V l, ], B d [V r,0 ] and B d [V r,0 ] are (d )-dimensional butterflies. We illustrate the above definitions in Figs. and 0, where as well as showing the decomposition of the butterflies B 5 and B into their constituent parts, we also detail two particular feedback vertex sets. We shall use these feedback vertex sets presently and consequently we name them as F B (B 5 ) and F B (B ), respectively. (Note that we split B in Fig. 0 into two halves, due to its size.) 0 B 5[V l,0] B 5[V l,] B 5[V r,0] B 5[V r,] 5 row V V V Figure. The butterfly B 5 with the feedback vertex set F B [B 5 ]. We leave it as an exercise for the reader to check that F B (B 5 ) and F B (B ) are indeed feedback vertex sets of B 5 and B, respectively. (Readers might find it instructive to first of all convince themselves that there are no cycles involving only vertices on two subsequent levels, and then to rule out potential cycles involving vertices on the bottom two levels, then cycles involving vertices on the penultimate and antepenultimate levels, and so on.) We are now in a position to detail our algorithm. Our algorithm outputs a feedback vertex set for B d which we denote F B [B d ], and we denote the feedback vertex sets of B d resulting from Algorithms A and L, in [], by F A [B d ] and F L [B d ], respectively (recall, Algorithm A is Chang, Lin and Lee s algorithm and Algorithm L is Luccio s algorithm, first derived in [5]). V 8

19 0 B [V ] l,0 B [V ] l, edges joining vertices on rows 5 and omitted from figure V V 5 row left-hand side of -dimensional butterfly B [V ] r,0 0 B [V ] r, edges joining vertices on rows 5 and omitted from figure V V 5 right-hand side of -dimensional butterfly row Figure 0. The butterfly B with the feedback vertex set F B [B ]. Algorithm B Input: The d-dimensional butterfly B d, where d 0. Output: The feedback vertex set F B (B d ) of B d. If d {0,,,,} then return F B (B d ) = F A (B d ) else if d = 5 then return F B (B 5 ) else if d = then return F B (B ) else return F B (B d ) = (Vd F B(B d [V l,0 ]) F L (B d [V l, ])) (Vd F L(B d [V r,0 ]) F B (B d [V r, ])).

20 That is, we proceed just as Chang, Lin and Lee did except the base cases of our recursive algorithm are different. The fact that our algorithm produces a feedback vertex set follows from the following lemmas from []. Lemma 0 [] For d >, suppose that F l,0 is a feedback vertex set of B d [V l,0 ] and that F l, = F L (B d [V l, ]). Then F l,0 F l, V d is a feedback vertex set of B d[v l ]. Lemma [] For d >, suppose that F r,0 = F L (B d [V r,0 ]) and that F r, is a feedback vertex set of B d [V r, ]. Then F r,0 F r, Vd is a feedback vertex set of B d [V r ]. Lemma [] For d >, suppose that F l Vd and that F r Vd are feedback vertex sets of B d [V l ] and B d [V r ], respectively. Then F l F r is a feedback vertex set of B d. Theorem The set F B (B d ) is a feedback vertex set of B d. Proof The proof follows from an elementary induction using the above lemmas. Not only can we use Chang, Lin and Lee s tools to prove that our algorithm is correct, we can also use their analysis to obtain the size of the feedback vertex set F B [B d ], for each d 5. From [], the size f A (d) of the feedback vertex set F A [B d ] is and (d + ) d + (d + ) d + d, if d is even, d d d +, if d is odd. Consequently, f A (5) = 50 and f A () = ; whereas, with f B (d) denoting the size of the feedback vertex set F B [B d ] produced by Algorithm B, f B (5) = 8 and f B () = 0. A simple observation yields that f A (7) f B (7) = ; f A (8) f B (8) = 8; f A () f B () = (f A (7) f B (7)) = 8; f A (0) f B (0) = (f A (8) f B (8)) = ; f A () f B () = (f A () f B ()) = ; f A () f B () = (f A (0) f B (0)) = ;... and a simple induction yields that f B (d) is equal to (d + ) d + d d, if d is even, and (d + ) d + Hence, the above are upper bounds on τ(b d ). d d d + d, if d 5 is odd. 0

21 5 Conclusion In this paper, we have improved the known upper and lower boards on the sizes of minimal feedback vertex sets in grids and butterflies. We feel that, subject to the resolution of the (rather annoying) class of grids one of whose dimensions is 5, the closeness of the resulting upper and lower boards should essentially close the investigation. The situation for butterflies is not so clear cut. Whilst we have managed to improve both upper and lower bonds, there is still some distance between the two bounds. We conjecture that the feedback vertex number for butterflies lies closer to our upper bound than our lower bound. Intuitively, we feel that our lower bound technique, which has only been applied at the extremities of the butterfly, should be applicable within the butterfly. Of course, we have so far been unable to do this. References [] I. Caragiannis, C. Kaklamanis and P. Kanellopoulos, New bounds on the size of the minimum feedback vertex set in meshes and butterflies, Information Processing Letters 8 (00) [] M-W. Chang, C-H. Lin and C-M. Lee, New upper bounds on feedback vertex numbers in butterflies, Information Processing Letters 0 (00) [] W.J. Dally and B. Towles, Principles and Practices of Interconnection Networks, Morgan Kaufmann (00). [] P. Festa, P.M. Pardalos and M.G.C. Resende, Feedback set problems, in: Handbook of Combinatorial Optimization (D.-Z. Du and P.M. Pardalos, eds.), Kluwer Academic Publishers, Supplement Vol. A (000) 0-5. [5] F.L. Luccio, Almost exact minimum feedback vertex sets in meshes and butterflies, Information Processing Letters (8) 5.