THE REVERSIBILITY OF GRAPH LAYOUT MULTISTRETCHABILITY Minko Markov Mathematical Linguistics Department, Institute of Mathematics and Informatics, BAS Acad. Georgi Bonchev St. Bl. 8, 1113 Sofia, Bulgaria [email protected]

Abstract We introduce the concepts of left- and right-extensibility of linear graph layouts with respect to a vertex set, and stretchability of linear graph layouts with respect to two vertex sets. We prove the stretchability does not depend on which of those vertex sets is associated with the left direction and which, with the right direction. Thus we extend a previous result saying that stretchability of layouts with respect to two vertices

Keywords graph theory, linear layouts, vertex separation.

1. INTRODUCTION Vertex Separation (VS) is an NP-Complete problem on graphs which is equivalent to many ostensibly distinct graph problems such as Pathwidth, Node Search Number, Interval Thickness, and others. Those can be thought of as different facets of the same computational problem which arises in diverse areas of knowledge such as Natural Language Processing (Kornai and Tuza (1992)), Graph Search Games (Kirousis and Papadimitriou (1985)), (Kirousis and Papadimitriou (1986)), VLSI Design (Lopez and Law (1980)) (Moehring 1990), Computational Biology (Goldberg and Golumbic and Kaplan and Shamir (1995)), Computational Complexity Theory (Robertson and Seymout (1983)), (Robertson and Seymour (1986)), and Network Reliability (Lucet and Manouvrier and Carlier (2000)). Since the problem is NP-Complete in general and remains so even for planar graphs with degree at most three (Monien and Sudborough (1988)), for chordal graphs (Gustedt (1993)), and for bipartite graphs (Goldberg and Golumbic and Kaplan and Shamir (1995)), one can hardly hope for any practical algorithm for VS on general graphs of non-trivial size. The motivation behind the main result from this paper is to develop algorithms for that problem for restricted graph classes, cactus graphs in particular. It is known (Bodlaender and Kloks (1996)) that for any graph with n vertices whose treewidth is bounded by a constant k there exists a polynomial time algorithm with time complexity Ω(n4k+3). Since cactus graphs have treewidth at most 2, it follows there is a polynomial time, Ω(n11) algorithm for their VS. However, that bound on the running time is not tight, the exponent 11 is too large to consider the algorithm practical, and, likely, there is a substantial hidden constant in the asymptotic running time expression. We would prefer an algorithm that is practically implementable and can run reasonably fast on cactus graphs with thousands of vertices, like the algorithm from (Ellis and Sudborough and Turner (1994)) for the VS of trees. The attempts to construct practical algorithms for the VS of restricted graph classes lead to the concept of what we call “extensibility with respect to a vertex”. Historically, that concept was first used by Skodinis (Skodinis (2000)) under the name “extendability” and independently discovered by Ellis and Markov (Ellis and Markov (2004)). Regardless of the term used for it, it is arises naturally because it allows us to construct bigger layouts out of smaller ones while maintaining a certain upper bound on the overall VS. The latter paper also defined “k-conforming layout with respect to two vertices” (ibid., Definition 3.3), a concept that was crucial for the construction of a fast, practical algorithm for the VS of unicyclic graphs. The extensibility of a layout allows us to grow it in one direction, either left or right, while the stretchability allows us to grow

it in both directions. The definition of k-conforming layout, however, has an imprecision in the sense that it does not always provide the intended properties. The problem with that definition and the way to remedy the situation is explained in (Markov (2007)). Thus the concept was given a new name, “stretchability”, which we use in the current work. An interesting property of the extensibility of a layout is that it is independent on the specific direction, left or right. In (Ellis and Markov (2004)) it is shown that for any graph there exists a k-left extensible with respect to any vertex u layout if and only if there exists a k-right extensible with respect to u layout. That discovery simplifies greatly the usage of extensibilities because it allows omitting the specific direction. The stretchability of a layout is independent on the specific directions, too. However, the proof of that claim in (ibid.) is faulty, because of the mentioned fault of the definition. A correct proof for the stretchability can be found in (Markov (2004)). Any cactus graphs with at least one cycle can be thought of as multitude of unicyclic graphs joined together in a tree-like way, so it was natural to try to construct an algorithm for the VS of cactus graphs, having the knowledge how deal with VS on trees and unicyclic graphs. While trying to construct such an algorithm (Markov (2007)), we discovered the necessity (ibid., Section 5) to be able to compute the stretchability of layouts with respect two sets of vertices, rather than with respect to two vertices. In the title of the current paper, we call that parameter “multistretchability”. Analogously to ordinary stretchability with respect to two vertices, the multistretchability of a layout does not depend on which vertex set is associated with the left direction and which, with the right one. That is the main result of the current work. We use it in our work in progress to derive the theoretical foundation of a fast, practical algorithm for the VS of cactus graphs.

2. PRELIMINARIES 2.1 Vertex Separation and Search Number We assume the reader is familiar with the basics of Graph Theory. A linear layout of a graph G = (V, E), or simply a layout, is a bijective function L: V → 1, 2,..., |V|. For any layout L and vertex u, π L (u) is defined to be π L (u) = {v ∈ V | L(v) ≤ L(u) and for some w ∈ V, L(w) > L(u) and (v, w) ∈ E}. The separation of u in L is |π L (u)|. For any v ∈ π L (u), we say that v contributes to the separation of u. The vertex separation of G under L is vs L (G) = max (|π L (u)|, such that u ∈ V) and the vertex separation of G is vs(G) = min (vs L (G) | L is a linear layout of G). Any layout of G with minimum separation is called optimal. For any u,v ∈ G such that L(u) < L(v), we say that under L, u is left of v, and v is right of u. Let w be the rightmost neighbour of u under L. The vertex right(u) is defined as follows: if L(w) > L(u), then right(u) is w, and otherwise right(u) is u. An interval in L is a possibly empty, contiguous subsequence of L. A node search on a graph G is a cleaning of the edges of G, performed according to the following rules. Initially, all the edges are contaminated. Then they are cleaned by the search in discrete subsequent moments by either placing a searcher on some vertex, or removing a searcher from some vertex that already had a searcher on it. There can be at most one searcher on any vertex at any moment. An edge becomes clean when both its endpoints have searchers on them. At any removal of a searcher from any vertex v, for any path p containing v such that p connects one endpoint of some clean edge e 1 with one endpoint of some contaminated edge e 2 , if v is the only vertex in p having a searcher prior to the removal, then e 1 becomes recontaminated. In other words, recontamination occurs infinitely fast, at any opportunity, and can only be prevented by the presence of searchers on every path between what has already been cleaned and what is still contaminated. The goal of the search is to clean all edges of G with as few searchers simultaneously used at any moment as possible. The smallest number of searchers that can perform the cleaning is the node search number of G. We do not introduce any other searching rules so we say shortly “search” and “search number” instead of “node search” and “node search number”. We denote the search number of G by sn(G).

It is obvious that during any search, every vertex must get a searcher at some moment. It is much less obvious that every search that can be done with k searchers, can be done with k searchers progressively which means that no vertex gets a searcher more than once (La Paugh (1993)), (Kirousis and Papadimitriou (1986)). That fact allows a more restricted definition of search. Suppose that G = (V, E) is any graph and |V| = n. A search S on G is a sequence of precisely 2n moves, two moves for each vertex: a positive move, which is associated with the placing of a searcher on that vertex, and a negative move, which is associated with the removal of that searcher, such that: 1. For each vertex, the positive move precedes the negative move. 2. The graph is cleaned according to the rules of node searching. For any vertex u ∈ V , we denote the positive move with u+, and the negative move with u−. If S is a search on some graph we define the reverse of S, denoted by rev(S), as the sequence of 2n moves that is the reverse permutation of S with the positive flags changed to negative and vice versa. It is obvious that the positive move precedes the negative move for each vertex in rev(S). Lemma 3.1 from (Ellis and Markov (2004)) proves that rev(S) fulfills the second condition for being a search, too, namely, that it cleans the graph, using the same number of searchers as S. Figure 1 illustrates the idea of search and the reversal of a search. Figure 1. G is a graph, S is a search on it, and rev(S) is the reverse of S

a

b g

f

e

G c

d

S = b+ a+ a− g+ b− d+ c+ c− d− f+ g− e+ f− e− rev(S) = e+ f+ e− g+ f− d+ c+ c− d− b+ g− a+ a− b−

For any graph G, vs(G) = sn(G) − 1 (Kirousis and Papadimitriou (1986)). That paper shows a deeper connection between those two parameters. First, for any optimal layout L of G, there is an optimal search S on G, such that the positive moves of S are in the same relative order as the order of the vertices in L, the negative moves are done as soon as possible (so that recontamination does not happen), and S uses vs L (G)+1, that is, vs(G)+1, searchers. We say that S is a search that corresponds to L. Second, for any optimal search S of G there is a unique optimal layout L of G such that the order of the vertices in L is the same as the order of the positive moves in S. We say that L is the layout that corresponds to S. That deeper connection allows us to say that Vertex Separation and Search Number are two facets of the same computational problem, and we may use whichever one of them we find more convenient. Suppose that L is an optimal layout for G and S is a search corresponding to L. By the said connection between Vertex Separation and Search Number, S is optimal. Suppose that L 1 is the layout corresponding to rev(S). By Lemma 3.1 from (Ellis and Markov (2004), rev(S) is a search using as many searchers as S, so rev(S) is an optimal search, and it follows that L 1 is an optimal layout of G. From the above discussion it is clear that L 1 is not uniquely determined by L. We say that L 1 is a reversal of L and we denote that by L 1 ∈ rev(L). Figure 2 illustrates distinct searches corresponding to the same layout and the non-uniqueness of layout reversals.

Figure 2. G is a graph, L is an optimal layout of G, both S 1 and S 2 are searches corresponding to L, and L 1 and L 2 are the layouts, corresponding to rev(S 1 ) and rev(S 2 ), respectively.

a

b g

f

e

G c L = S1 = S2 = rev(S 1 ) = rev(S 2 ) = L1 = L2 =

d bagdcfe b+ a+ a− g+ b− d+ c+ c− d− f+ g− e+ f− e− b+ a+ a− g+ b− d+ c+ d− c− f+ g− e+ e− f− e+ f+ e− g+ f− d+ c+ c− d− b+ g− a+ a− b− f+ e+ e− g+ f− c+ d+ c− d− b+ g− a+ a− b− efgdcba fegcdba

2.2 Extensibility and Stretchability The following two definitions assume that G = (V, E) is a graph, L is a layout of it, U ⊆ V, where U = {u 1 , u 2 , …, u m }, and k is a positive integer. Definition 1 (left extensibility with respect to multiple vertices). Assume that the relative order from left to right of the vertices from U in L is precisely u 1 , u 2 , …, u m . We say that L is left-extensible with respect to k and U if all of the following hold: • All vertices right of and including u m have separation ≤ k. • All vertices left of u m have separation ≤ k−1. • All vertices left of u m-1 have separation ≤ k−2. • … • All vertices left of u i have separation ≤ k−m+i−1. • … • All vertices left of u 2 have separation ≤ k−m+1. • All vertices left of u 1 have separation ≤ k−m. That is denoted by “L is lext(k,U)”. ■ Observation 1. If we add m vertices not from G to the left side of L and connect each one of them to a distinct vertex from U and call the thus obtained layout L*, then L is left extensible with respect to k and U if and only if the separation of the new layout L* is at most k. ■ Clearly, if vs L (G) ≤ k and L starts with any permutation of the vertices from U, then L is lext(k,U). On the other hand, it is true that if L is lext(k,U), then the vertices from U can be moved, in whatever order, to the leftmost m positions and L will remain lext(k,U). However, the latter fact is not immediately obvious, so we prove it rigorously in Corollary 1. That corollary allows us, whenever we have a layout that is lext(k,U), to think without loss of generality that it starts with the vertices from U. Definition 2 (right extensibility with respect to multiple vertices). Suppose that w i = right(u i ) for 1 ≤ i ≤ m, that those w i are not necessarily distinct, and that W = U mi=1 wi . Assume that the relative order

from left to right of the vertices from W in L is precisely w 1 , w 2 , …, w m . We say that L is right-extensible with respect to k and U if all of the following hold: • All vertices left of w 1 have separation ≤ k. • All vertices right of and including w 1 have separation ≤ k−1. • All vertices right of and including w 2 have separation ≤ k−2. • … • All vertices right of and including w i have separation ≤ k−i. • … • All vertices right of and including w m-1 have separation ≤ k−m+1. • All vertices right of and including w m have separation ≤ k−m. That is denoted by “L is rext(k,U)”. Note that the order of the vertices from U in this definition is immaterial. ■ Observation 2. If we add one or more vertices not from G to the right side of L and connect each vertex from U to one or more of them and call the thus obtained layout L*, then L is right extensible with respect to k and U if and only if the separation of the new layout L* is at most k. ■ Definition 3 (stretchability). Suppose that G = (V, E) is a graph and U and W are not necessarily disjoint subsets of V, each of them being non-empty or empty. Say, U = {u 1 , u 2 ,… ,u n } and W = {w 1 , w 2 , …, w m }. Suppose that L is a layout of G and the leftmost and the rightmost vertices of L are a and z, respectively. Suppose that ℐ i,left is the possibly empty interval in L from and including a to and excluding u i , for 1 ≤ i ≤n.

Suppose that ℐ j,right is the possibly empty interval in L from and including right(w j ) to and including z, for 1 ≤ j ≤ m. Suppose that ℑ i,left is the possibly empty interval in L from and including a to and excluding w i , for 1

≤ i ≤ m. Suppose that ℑ j,right is the possibly empty interval in L from and including right(u j ) to and including z, for 1 ≤ j ≤ n. We say that L is k-stretchable with respect to U and W if at least one of the following holds: 1.

The separation of any vertex in L is at most k minus the number of intervals from ℐ 1,left , ℐ 2,left , …,

ℐ n,left and ℐ 1,right , ℐ 2,right , …, ℐ m,right that it is in.

2.

The separation of any vertex in L is at most k minus the number of intervals from ℑ 1,left , ℑ 2,left , …,

ℑ m,left and ℑ 1,right , ℑ 2,right , …, ℑ n,right that it is in. In the former case, we say that U is associated with the left direction and W, with the right one. In the latter case, we say the opposite. ■ Observation 3. Since the separation of any vertex in L is non-negative, it follows that k ≥ max(m, n). ■

Our Theorem 1 from the next section says that the first condition in Definition 3 implies the second condition, and vice versa. In other words, the definition has a redundancy. Note that if one of U, W in Definition 3 is empty then the definition becomes equivalent to Definition 1 or Definition 2. It follows that extensibility is a special case of stretchability. Observation 4. In the naming convention of Definition 3, if we add n vertices not from G to the left side of L and connect each one of them to a distinct vertex from U and we add one or more extra vertices not from G to the right side of L and connect each vertex from W to one or more of them, and we call the thus obtained layout L*, then L is k-stretchable with respect to U and W so that U is associated with the left direction and W, with the right direction, if and only if the separation of the new layout L* is at most k. ■

3. LEMMAS AND THEOREMS Lemma 1. Suppose that G = (V, E) is a connected graph. Suppose that W and Z are two not necessarily disjoint vertex subsets of V and L is a layout for G that is k-stretchable with respect to W and Z where W is associated with the left direction and Z, with the right one. Then G has a layout L 1 that starts with some ordering of the vertices of W, the remaining vertices having the relative order as in L, and L 1 is rext(k, Z).

Proof: Suppose that W ∩ Z = B, B ≠ φ , and u ∈ B . Call L u the layout for G − u that is obtained from L by deleting u. It is easy to see that L being k-stretchable with respect to W and Z, regardless of the directions that W and Z are associated with, is equivalent to L u being (k−1)-stretchable with respect to W−u and Z−u, provided the directions associated with W and Z are the same. If we generalise that observation we reduce the current lemma to the following equivalent claim, in which G B is G with the vertices from B deleted and b = |B|. Suppose that G B has a layout L B that is (k−b)-stretchable with respect to W \ B and Z \ B, where W \ B is associated with the left direction and Z \ B, with the right direction. Then G B has a layout L 2 that starts with some ordering of the vertices from W \ B, the remaining vertices have the same relative order as in L, and L 2 is rext(k-b, Z \ B). Therefore, we can assume without loss of generality that W ∩ Z = φ .

Suppose that W = {w 1 , w 2 , …, w q } and Z = {z 1 , z 2 , …, z t }. Assume that x i = right(z i ), for 1 ≤ i ≤ t , and X is the union of all x i . Now move the vertices from W to the leftmost q positions of the layout, leaving the relative order of the remaining vertices unchanged. Call the obtained layout, L 1 . We claim that L 1 is rext(k,Z). First consider the “starting sequence” of L 1 which consists of the vertices from W. Obviously, the maximum separation of any vertex in it is q. We already pointed out (Observation 3) that q ≤ k and so all the w i ’s conform to the requirement of Definition 2 to have separation at most k. Now consider the remaining vertices of L 1 . For any vertex v from V \ W, define the following two functions. Let f(v) be the number of vertices from W that are right of v in L. Let g(v) be the number of vertices from X that are left of, or coinciding with, v in L. Definition 3 implies that | π L ( v ) |≤ k − ( f ( v ) + g ( v )) . After we move the vertices from W to the left side of the layout, the separation of v may increase by at most f(v), since that is the number of vertices from W that could not possibly contribute to the separation of v in L, being right of it in L, but can possibly contribute to the separation of v in L 1 , being left of it in L 1 . It follows that | π L1 ( v ) |≤ k − g ( v ) . But that is precisely the requirement of Definition 2 for right extensibility with respect to k and Z. ■ Lemma 1 allows us, being given a layout that is k-stretchable with respect to two vertex subsets, to think without loss of generality that it starts with the vertices of one of them and is right extensible with respect to k and the other vertex subset. Corollary 1. Suppose that G = (V, E) is a connected graph, W = {w 1 , w 2 , …, w q } is a subset of V, and L is a layout for G that is lext(k, W). Then there is a layout L 1 for G such that L 1 starts with some ordering of the vertices from W, the remaining vertices in L 1 have the same relative order as in L, and vsL1 (G) ≤ k. Proof: Apply Lemma 1 with Z = φ . ■

Lemma 2 (Ellis and Markov (2004)). Suppose that G is a connected graph and S is a search on it. Then rev(S) is a search on G as well. Furthermore, rev(S) uses the same number of searchers as S. ■ Corollary 2 (Ellis and Markov (2004). Suppose that G is a connected graph and L is a layout for G. Suppose that L 1 ∈ rev(L). Then vs L (G) ≤ k implies vs L1 (G) ≤ k and vice versa. ■ Theorem 1. Suppose that G = (V, E) is a connected graph. Suppose that W and Z are two not necessarily disjoint vertex subsets of V. Then G has a layout that is k-stretchable with respect to W and Z where W is associated with the left direction and Z with the right direction, if and only if G has a layout that is kstretchable with respect to W and Z where W is associated with the right direction and Z is associated with the left direction.

Proof: Suppose that W = {w 1 , w 2 , …, w q } and Z = {z 1 , z 2 , …, z t }. Suppose that L is a layout for G that is kstretchable with respect to W and Z where W is associated with the left direction and Z with the right direction. By Lemma 1 we assume without loss of generality that L starts with the vertices w 1 , w 2 , …, w q , in that order and L is rext(k, Z).

Suppose that A = {a 1 , a 2 , …, a q } and B = {b 1 , b 2 , …, b t } are disjoint sets of vertices that are disjoint with V as well. Suppose that G 1 is the graph obtained from G by adding the vertices from A and B, joining a i and w i with an edge for 1 ≤ i ≤ q, and joining b i and z i with an edge for 1 ≤ i ≤ t. Suppose that L 1 is the layout for G 1 such that L 1 = a 1 , a 2 , …, a q , L, b 1 , b 2 , …, b t . By Observation 4, we know that vs L1 (G 1 ) ≤ k. As L starts with the vertices w 1 , w 2 , …, w q in that order, we have: L 1 = a 1 , a 2 , …, a q , w 1 , w 2 , …, w q , ………… b 1 , b 2 , …, b t Suppose that S 1 is a search corresponding to L 1 . By Theorem 4.1 of (Kirousis and Papadimitriou (1986)), S 1 uses at most k+1 searchers. In general, S 1 has the form: S 1 = a 1 +, a 2 +, …, a q +, w 1 +, …, w 2 +, …, w q +, …, b 1 +, …, b 2 +, …, b t +, …… where the sequence a 1 +, a 2 +, …, a q + consists only of positive moves for the a i ’s. We may assume without loss of generality that S 1 has the form: S 1 = a 1 +, a 2 +, …, a q +, w 1 +, a 1 −, …, w q +, a q −, ……, b 1 +, z 1 −, b 1 −, …, b t +, z t −, b t − where the sequence a 1 +, a 2 +, …, a q + consists only of positive moves for the a i ’s, the sequence w 1 +, a 1 −, …, w q +, a q − consists only of positive moves for the w i ’s, each one of them immediately followed by negative move for the corresponding a i , and the sequence b 1 +, z 1 −, b 1 −, …, b t +, z t −, b t − consists only of repeating triples b i +, z i −, b i −. Now consider rev(S 1 ). By Lemma 2, rev(S 1 ) is a search too, using at most k+1 searchers. Clearly, rev(S 1 ) has the form: rev(S 1 ) = b t +, z t +, b t −, …, b 1 +, z 1 +, b 1 −, ……, a q +, w q −, …, a 1 +, w 1 −, a q −, …., a 1 − Suppose that rev(L 1 ) is the layout for G 1 corresponding to rev(S 1 ). By the cited result of Kirousis and Papadimitriou, we know that vs L1 (G 1 ) ≤ k. It is clear that rev(L 1 ) has the form: rev(L 1 ) = b t , z t , b t−1 , z t−1 , ..., b 1 , z 1 , ……, a q , a q−1 , …, a 1 Now consider the layout L 1 * for G 1 which is obtained from rev(L 1 ) by moving the vertices z t , z t−1 , …, z 1 , in that order, immediately to the right of b 1 and leaving the other vertices intact: L 1 * = b t , b t−1 , …, b 1 , z t , z t−1 , …, z 1 , ……, a q , a q−1 , …, a 1 It is clear that vs L* (G 1 ) ≤ k, because by the premises k ≥ t. Suppose that L* is the layout for G obtained from 1

L 1 * by deleting the vertices a 1 , a 2 , …, a q and b 1 , b 2 , …, b t . Thus, L 1 * = b t , b t−1 , …, b 1 , L*, a q , a q−1 , …, a 1 . Apply Observation 4 and conclude that L* is a k-stretchable with respect to W and Z layout for G, such that Z is associated with the left direction and W is associated with the right direction. ■

4. CONCLUSION The symmetry with respect to the left and right directions of k-stretchability with respect to two vertex sets is not surprising, given our previous knowledge of layout reversibility and reversibility of stretchability with respect to two vertices. Although the result is intuitively clear, its proof is not trivial. Once proved, it is quite valuable when we work with stretchable with respect to two vertex sets layouts because it allows us to focus only on the vertex sets with respect to which we stretch, rather than on the vertex sets and the direction associated with each set.

REFERENCES Bodlaender, H. and T. Kloks (1996). Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs. Journal of Algorithms, Vol. 21, No. 2, pp. 358-402. Ellis, J. and M. Markov (2004). Computing the Vertex Separation of Unicyclic Graphs. Information and Computation, Vol. 192, pp. 123-161.

Ellis, J. and I. Sudborough and J. Turner (1994). The Vertex Separation and Search Number of a Graph. Information and Computation, Vol. 113, No. 1, pp. 50-79. Goldberg, P. and M. Golumbic and H. Kaplan and R. Shamir (1995). Four strikes against physical mapping of DNA. Journal of Computational Biology, Vol. 2, pp. 139-152. Gustedt, J. (1993). On the Pathwidth of Chordal Graphs, Discrete Applied Mathematics. Vol. 45, pp. 233-248. Kirousis, L. and C. H. Papadimitriou (1985). Interval Graphs and Searching. Discrete Mathematics, Vol. 55, pp. 181-184. Kirousis, L. and C. H. Papadimitriou (1986). Searching and pebbling. Theoretical Computer Science, Vol. 47, No. 2, pp. 205-218. Kornai, A., Z. Tuza (1992). Narrowness, pathwidth, and their application in natural language processing. Discrete Applied Mathematics, Vol. 36, pp. 87-92. LaPaugh, A. (1993). Recontamination Does Not Help to Search a Graph. Journal of the ACM, Vol. 40, No. 2, pp. 224245. Lopez, A. and H. Law (1980). A dense gate matrix layout method for MOS VLSI. In IEEE Transactions on Electronic Devices, No. 27, pp. 1671-1675 Lucet and Manouvrier and Carlier (2000). Evaluating Network Reliability and 2-Edge-Connected Reliability in Linear Time for Bounded Pathwidth Graphs. Algorithmica, Vol. 27, pp. 316-336. Markov, M. (2004). A Fast Practical Algorithm for the Vertex Separation of Uncyclic Graphs. Master Thesis, University of Victoria, Victoria, British Columbia, Canada, http://minko.m.markov.googlepages.com/diss.pdf. Markov, M. (2007). On the Vertex Separation of Cactus Graphs. Serdica Journal of Computing, Vol.1, pp. 45-72. Moehring (1990). Graph Problems Related to Gate Matrix Layout and PLA Folding. Computational Graph Theory, Vol. 7, pp. 17-51. Monien, B. and I. H. Sudborough (1988). Min Cut is NP-complete for Edge Weighted Trees. Theoretical Computer Science, Vol. 58, No. 1-3, pp. 209-229. Robertson, N. and P. D. Seymour (1983). Graph Minors I, Excluding a Forest. Journal of Combinatorial Theory Series B, Vol. 35, pp. 39-61. Robertson, N. and P. D. Seymour (1986). Graph Minors II: Algorithmic Aspects of Tree-width. Journal of Algorithms, Vol. 7, pp. 309-322. Skodinis, K. (2000). Computing optimal strategies for trees in linear time. Proceedings of the 8th Annual European Symposium on Algorithms. Saarbruecken, Germany, pp. 403-414.