On vertex ranking for permutation and other graphs

Citation for published version (APA):Deogun, J. S., Kloks, A. J. J., Kratsch, D., & Müller, H. (1993). On vertex ranking for permutation and othergraphs. (Computing science notes; Vol. 9330). Technische Universiteit Eindhoven.

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Eindhoven University of Technology

Department of Mathematics and Computing Science

On Vertex Ranking for Permutation and other Graphs

by

1.S. Deogun T. Kloks D. Kratsch H. MUller

93/30

Computing Science Note 93/29 Eindhoven, September 1993

On Vertex Ranking f1r Permutation and Other , Graphs

I .J.S·IDeogun •

Department of Compu,ter Science and Engineering University of Nebraska - Lincoln

Lincoln, NE!68588-0ll5, USA

T. Kloks t

Department of Mathemlatics and Computing Science Eindhoven Unitersity of Technology

I P.OI.Box 513

5600 MB Eindhren, The Netherlands

D. Kratsch II and H. Miiller §

Fakultat fiil' MatHematik und Infol'matik Fl'iedl'ich-Sdhiller-U nivel'sitat ,

U niversitatshochhaus I

07740 J'ina, Germany

, i

Abstract

An optimal vert,ex ranking br a permutation graph can be com­puted in time O(n6), where n ik the number of vertices. The demon­strated method call be used for Idesigning polynomial time algorithms computing an optimal vertex ranking on the following classes of well­structured graphs t.oo: circular I permutation graphs, interval graphs, circular arc graphs, trapezoid graphs and cocomparability graphs of bounded dimension. !

1 Introduction

In this paper we present a polynomiial time algorithm which solves the vertex ranking problem 011 permuta.tion haphs. The vertex ranking problem is also called the oNlered colori1l9 problem [15J. The problem has received mnch attention lately because of the growing number of applications. For example, the problem of finding an hptimal vertex ranking is equivalent with the problem of finding the minimUlri height elimination tree of a graph. This , measure is ofimportallce in computing Cholesky factorizations of matrices in

I -deogunGcse.unl,edu i ttontvin.tue.nl i lDIETER.KRATSCHG.atheaatik,uni-jena.dbp.de §HAIKO.KUELLERGaatheaatik,uni-jerla.dbp.de

I

1

parallel [19, 9, 4J. Other applications lie in the field of VLSI-layout [18, 23J. Yet other applications can be found in scheduling problems of assembly steps in manufacturing systems [13, 25, 14,20, 22J.

Much work has been done in finding optimal rankings of trees. For trees there is now a linear time algorithm finding an optimal vertex ranking [21J. For the closely related edge ranking problem on trees an O( n3 ) algorithm is given in [8J. Recently, the polynomial time solvability was rediscovered in [25J, however the time complexity is O( n3 10g n). Efficient vertex ranking algorithms were known for very few other classes of graphs . The vertex ranking problem is trivial on split graphs and can also be solved in linear time on cographs [22J. Very recently, we heard about an 0(n4) algorithm for vertex ranking of interval graphs [IJ (however, we have not seen the paper yet).

The decision problem 'Given a graph G and a positive integer k, has G a vertex ranking with at most k colors' is NP-complete, even when restricted to cobipartite or bipartite graphs [3J. In view of this it is interesting to notice that for each constant t, the class of graphs with vertex ranking number at most t is recognizable in linear time [3J.

In [1.5J, among other things, an O(,jii) bound is given for the vertex ranking number of a planar graph and the authors describe a polynomial time algorithm which finds a ranking using only O(,jii) colors. For graphs in general there is an approximation algorithm known with factor O(log2 n) [4, 16J. In [4J it is a.lso shown that one plus the pathwirith of a graph is a lower bound for the vertex ranking number of the graph (hence a planar graph has pathwidth O( ,jii), which is also shown in [16J using different methods).

One of the most well-known and well-studied classes of perfect graphs is the class of 1Jermutalio71 graphs. Permutation graphs are exactly the comparability graphs of pasels of dimension at most two. They can also be characterized as the graphs which are at the same time a comparability and a cocomparability graph. It follows that there is also a characterization in terms of forbidden induced subgraphs [11, 2J. Many problems such as DOMINATION, CLIQUE, INDEPENDENT SET, TREEWIDTH and MINIMUM FILL­

IN can be solved efficiently for permutation graphs [10, 6, 26, 5, 17]. For definitions and properties of classes of well-structured graphs not given here we refer to [12, 6, 2J.

In this paper we show that the vertex ranking problem can be solved efficiently for permutation graphs. Furthermore, this approach can be used to design polynomial time algorithms solving the vertex ranking problem also on circular permutation graphs, interval graphs, circular arc graphs, trapezoid graphs and cocomparahility graphs of hounded dimension.

2 Preliminaries

We start with some definitions and easy lemmas on vertex rankings and on permutation graphs. We start with some preliminaries on rankings.

2

2.1 Preliminaries on rankings

Definition 2.1 Let G = (V, E) be a graph and let t be some integer. A (vertex) t-ranking is a coloring c: V ---> {l, ... , t} such that for every pair of vertices x and y with c( x) = c(y) and for every path between x and y there is a vertex z on this path with c(z) > c(x). The rank of G, X,(G), is the smallest value t for which the graph admits at-ranking.

Clearly, a vertex ranking is a proper coloring. Hence X,(G) 2: X(G) for every graph G. We call a x,(G)-ranking of G an optimal ranking.

Lemma 2.1 Let G = (V, E) be connected, and let c be a t-ranking of G. Then there is at most one vertex x with c( x) = t.

Proof. Assume there are two vertices with color t. Since G is connected, there is a path between these two vertices. By definition this path must contain a vertex with color at least t + 1. This is a contradiction. 0

Remark 2.1 Notice that if c is a t·ranking of a graph G and H is a subgraph of G, then the restriction of c to the vertices of H is a t-ranking for H.

This observation together with Lemma 2.1 leads to. the following lemma which appears in [15J. If G = (V, E) is a graph and X <;; V is a subset of vertices, then we denote by G[XJ the subgraph of G induced by X.

Lemma 2.2 A function c: V ---> {I, .. . ,t} is at-vertex ranking for a graph G = (V, E) if and only if for each 1 :s i :s t, each connected component of the subgraph G[{x I c(x) ~ i}J olG has at most one vertex y with cry) = i.

A concept called elimination tree, or separator tree, is closely related to ranking.

Definition 2.2 Let G = (V, E) be a connected graph. An elimination tree for G is a rooted tree T with vertex set V defined recursively as follows. If V = {x} then T is the rooted tree containing only one vertex x. Otherwise choose a vertex rEV as the root of T. Let Clo ... , Cp be the connected components of G[V \ {r} J. For each component Ci let Ti be an elimination tree. T is defined by making each root ri of Ti adjacent to r.

The height of a rooted tree T is the length of a longest path from the root to a leaf. The following result appears, in different form, also in [3J.

Lemma 2.3 Let G be connected. Let h(G) be the smallest height of an elimination tree. Then X,(G) = h(G) + 1.

Proof. Consider a x,(G)-ranking c of G. We show that there is an elimina­tion tree with height at most X,(G) - 1. If G has only one vertex, this is obvious. Otherwise, there is exactly one vertex with color X,( G). Make this vertex the root r of T. For each connected component of G[V \ {r}J, the

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restriction of c is a (x,(G) - I)-ranking. By induction, for each connected component there is an elimination tree of height at most Xr(G) - 2.

Now let T be an elimination tree of height h. Make a coloring of G by coloring the vertices level by level: Color the root (at level h + 1) with color h + 1. All vertices of level i (1 :<::: i :<::: h + 1) get color i. Now let x and y be two vertices at the same level. Consider the set 5 of vertices at higher levels. By definition of an elimination tree, x and yare in different connected components of G[V \ 5J. Since 5 contains only vertices with larger colors, every path between x and y must have a vertex of larger color. 0

Remark 2.2 Given an elimination tree with height h, a ranking can be obtained by COl07'iug the vedice. according to the level in which they appear in T, where the mot has the highest level.

Remark 2.3 Using results of !41 it follows that there exists an O(log2 n) approximation algorithm to determine the vertex ranking number of a graph.

Definition 2.3 A subset 5 ~ V is an a, b-separator for nonadjacent ver­tices a and b, if the removal of 5 separates a and b in distinct connected components. If no proper subset of 5 is an a, b-separator then 5 is a min­imal a, b-separator. A minimal separator 5 is a subset such that 5 is a minimal a, b-sepamtor for some nonadjacent vertices a and b.

The following theorem is our main tool for designing efficient vertex ranking algorithms on permutation and other graphs.

Theorem 2.1 Let G = (V, E) be a graph which is not a clique. Then

Xr(G) = min max (Xr(C) + 15i), s C

where 5 is a minimal vertex separator in G and C is II connected component

ofG[V \ 5J.

Proof. We first show that Xr(G) :<::: maxcXr(C) + 151 for all minimal sep­arators 8. We may assume that the graph is connected. First let 8 be minimal separator and let R be the maximum of the ranking number over all components. Create an elimination tree with height R+ 181-1 as follows_ First choose all vertices of 8. Clearly this takes at most 151 levels. Each component has an elimination tree with height at most R - 1, hence the total height of the elimination tree is at most R + 151 - 1.

Let T be an elimination tree of height Xr(G)-1 (i.e., of minimum height), and consider the coloring c given by the levels (Remark 2.2). Let Xr(G) - s be the highest level in the tree containing more than one vertex. Let 5 be the set of vertices at higher levels. Hence 181 = s. Notice that s < [VI since G is not a clique. Then G[V \ 5J is disconnected. Hence there is a minimal separator 8' ~ 5. The coloring c is such that all vertices of 8 have a unique color from {X(G), ... ,X(G) - s + I}. Obviously, we may permute these colors such that the vertices of 8' all have a unique color from

4

{x(G), ... ,x(G) - IS'I + I}. Then all vertices of G[V \ S'] have colors of {I, .. ',X(G) -15'1}, and hence each component has vertex ranking number at most X(G) -IS'I. 0

Remark 2.4 Notice that if S is a minimal separator such that all compo­nentsC have vertex ranking number at most x,(C)-ISI, then there is a rank­ing such that all vertices of S have a unique color from {x,( G), ... , x,( G)­lSI + I} and all other vertices have a smaller color.

Remark 2.5 The formula in Theorem 2.1 holds just as well if we replace 'minimal separator' by 'separator' or by 'inclusion minimal separator' (i.e., a minimal separator that is not properly contained in any other minimal separator).

2.2 Preliminaries on permutation graphs

A permutation 11" of the numbers l, ... ,n is a sequence 11" = {7r}, ... ,7rnJ.

Definition 2.4 If 11' is a l'cnnutation of the numbers I, ... , n, we can con-struct a graph G[rr J = (V, E) with vertex set V = {1, ... , n} and edge set E:

(i,j) E E {o} (i - j)(rr;-I - 11';1) < 0

A n undirected graph is a permntation graph if there is a permutation 11' such that G"., G[rrJ.

The graph G[rrJ is sometimes called the inversion graph of 11'. If the permu­tation is not given, it can be computed in O(n2 ) time [12, 24J. In this paper we assume that the permutation is given and we identify the permutation graph with the inversion graph. A permutation graph is an intersection graph, which is illustrated by the matching diagram. These notions appear for example in [12J.

Definition 2.5 Let 11' be a permutation of 1, ... , n. The matching diagram can be obtained as follows. Write the numbers 1, ... , n horizontally from left to right. Underneath, write the numbers 11'1, ••• ,11',,, also horizontally from left to right. Draw straight line segments joining the two 1 's, the two 2 's, etc.

Notice that two vertices i and j of G[rrJ are adjacent if and only if the corresponding line segments intersect. This is illustrated in figure 1. we give an example.

Definition 2.6 A scanline in the diagram is any line segment with one end vertex on each horizontal line, such that the end points of the scanline do not coincide with end points of other line segments in the diagram. A scanline .5 is between two non crossing line segments x and y if the top point of s is in the open interval between the top points of x and y and the bottom point of s is in the open inte"val between the bottom points of x and y.

5

5 1 1 234 5

4<0 2 3 ~ 3 5 142

Figure 1: permutation graph and matching diagram

Consider two nonadjacent vertices x and y. The line segments in the diagram corresponding to x and y do not cross. Hence we can find a scanline s between the lines x and y. Take out all the lines that cross the scanline s. Clearly this corresponds to an x, v-separator in the graph. The next lemma, which appears in [5J, shows that we can find all minimal x, v-separators in this way.

Lemma 2.4 Let G be a permutation graph, and let x and y be nonadjacent vertices in G. Every minimal X, y-separalor consists of all line segments crossing a scanline which lies between the line segments of x and y.

If s is a scanline, then we denote by S the set of vertices of which the corre­sponding line segments cross s. We call two scan lines 81 and 82 equivalent, 8, :; 82, if they have the same position in the diagram relative to every line segment; i.e., the set of line segments with the top (or bottom) end point to the left of the top (or bottom) end point of the scanline is the same for 8,

and 82.

Corollary 2.1 There (Lre O(n2) minimal separators in a permutation graph with n vertices.

Before we can state and prove our main theorem in the next section, we need a few more definitions.

Definition 2.7 Let 8, and S2 be two scanlines of which the intersection is either empty or one of the end points of 8, and S2' A piece C = C(SIo S2) is a subgraph of G induced by the following 8ets of lines:

• All lines that are oetween the scanlines (in case the scanlines have a common end point, this set is empty) .

• All lines crossing at least one of the scanlines.

We identify the piece C = C(S"'2) with the diagram containing SI, S2 and the set of lines corresponding with vertices of C.

Definition 2.8 Let C = C(", S2) be a piece. A scanline t is splitting if the top point of t is in the closed interval between the top points of Sl and S2,

and the bottom point of t is in the closed interval between the bottom points of s, and S2'

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3 Feasible pieces

Definition 3.1 Let C = C(sJ, S2) be a piece. C is called t-feasible if either

1. C has at most t vertices, or

2. there is a t-ranking of C such that each vertex of 51 U 52 has a unique color from {t,t - 1, ... ,t -151 U 521 + I} and all other vertices ofC have a color in {I, 2, ... , t -151 U 521}.

Theorem 3.1 Let C = C(SIoS2) be a piece with at least t+ 1 vertices. Then C is t-feasible if and only if there is a splitting scanline s such that the two pieces C1 = C( SI, s) and C2 = C( s, S2) are both smaller than C and Co is q;-feasible (i = 1,2), where 'J; = t -151 U .'h U 51 + 15; U 51.

Proof. First assume that C is I-feasible. Then there is a t-ranking such that the vertices of 51 U 52 all have unique colors from {I, ... , t - 151 U 521 + I} and all other vertices have smaller colors. Since the number of vertices is at least t + 1 there must be nonadjacent vertices x and y which have the same color. Then x and y cannot be elements of 51 U 52. Let C' be the induced subgraph of C arising by the removal of the vertices of 51 U 52' Then C' has vertex ranking number at most t' = t - 151 U 521 and C' is not a clique.

By Theorem 2.1, there exists a minimal separator 5' in C' such that all components have vertex ranking number at most t' - 15'1. Let a and b be nonadjacent vertices such that 5' is a minimal a, b-separator in C'. By Remark 2.4, C' has a ranking such that all vertices of 5' have a unique color from {t', t' - 1, ... , t' - IS'I + I} and all other vertices have smaller colors.

Consider the diagram for C', i.e. the diagram for C with line segments of 8 1 U 52 removed. By Lemma 2.4 there exists a scanline s between a and b such that vertices of S' correspond exactly with the line segments that cross s. Notice that since a and b are not elements of 51 U 52, the scanline s must be splitting in C. Notice also that S contains only vertices of .'h U 52 U 5'. This shows that there is a ranking of C such that all vertices of 51 U 52 U 5 have a unique color from {t - lSI U 52 U SI + 1, t - 151 U S2 U 51 + 2, ... , t} and all other vertices have colors from {I, ... , t - 151 U 52 U 51}. It follows that C; is smaller than C and qi-feasible (i = 1,2).

Now assume there is a splitting scanline s such that C1 and C2 are smaller and Ci is qi-feasible (i = 1,2). Then there is a coloring of C such that the vertices of C not belonging t.o S I U 8 2 U S have colors from {1 , 2, ... , t -151 U 52 U 51}. Hence we can find a ranking by assigning a unique color from the set {t - 151 U 52 U 51 + 1, t - lSI U 52 U 51 + 2, ... , t} to the vertices of ~U~U5. 0

4 Computing the vertex ranking number of a permutation graph

In this section we describe an aIgorithm to compute the vertex ranking number of a permutation graph. The first step of the algorithm computes all

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different pieces and sorts them according to increasing number of vertices. For each piece in turn (starting with the components with the smallest number of vertices) we compute the smallest t such that the component C is t-feasible. This is stored as t( C). We can determine whether the piece is t-feasible using Theorem 3.1. Notice that the largest piece has a scanline SL

which lies totally to the left of all line segments and a scanline SR which lies totally to the right of all line segments. In other words, this piece is just the graph G itself. Notice also that this piece is t-feasible if and only if there is a ranking of G with t colors, since SI = S2 = 0.

Theorem 4.1 There is a O( nS) algorithm computing the vertex ranking number of a permutation graph.

Proof. The existence of the algorithm is clear by the discussion above. The number of pieces is bounded by (n + 1)4 since it is determined by two scanlines. The test whether a piece C = C(st. S2) is t-feasible can be performed by trying all possible scanlines 8 between SI and 82, testing for each if the condition of Theorem 3.1 is satisfied. The number of these scanlines is at most (n + 1)2.

Now let C = C(ShS2) be a piece and let S be a splitting scanline. For computing the smallest t such that C is t-feasible via CI = C(s!, s) and C2 = C( 8,82) we simply need the following anxiliary data: a( s, s2)the number of line segments crossing S2 but not s (and stl, and b(8!, s)the number of line segments crossing Sl but not 8 (and 82). Notice that a(s,82) = IS2 \ SI = IS2 \ (SU SI)I and b(SI' 5) = lSI \ SI = 1.5't \ (SU S2)1 hold, using the notation of Theorem 3.1.

Assume we had computed all the auxiliary data in a preprocessing. Then, we get the smallest t such that a piece Cis t-feasible via CI = (SI'S) and C2 = (8, S2) by the formula:

t.( C) = ma.x( t( Cd + a( s, S2), t( C2) + b( SI, s)).

Consequently, t( C) = min, t,( C) where S is a splitting scanline of C = C(SI' S2).

Computing all the 0(n4) auxiliary data a(r,s) and b(r,8) for scanlines rand s having at most one endpoint in common can easily be done within the overall time bound. Moreover, a standard approach for computing such values using the matching diagram will do the preprocessing in time 0(n4) (see e.g. [5]). 0

Remark 4.1 An optimal vertex ranking of a given permutation graph can also be computed in time O(nS). Whenever t(C) has been computed by table look-up to smaller pieces we add pointers to a Imir giving raise to the value t(C). After finding X,(G) a backtracking using these pointers will produce an optimal vertex ranking or a minimum elimination height tree of G.

8

5 Other well-structured graphs

We described a simple algorithm to compute the vertex ranking number of a permutation graph.

The demonstrated 'piece approach' does not rely much on the structure of permutation graphs. Indeed, the key properties a graph class should have for using this approach are:

1. the number of minimal separators of a graph is bounded by a polyno­mial in n, and

2. the number of pieces of a graph is bounded by a polynomial in n.

In general a piece of a graph G is an induced sub graph C of G created as follows: Choose an arbitrary subset S of all minimal separators of G. Choose an arbitrary connected component of G[V \ USESS] and all minimal separators in S having a neighbor in the component. This is the vertex set of a piece.

If there is a polynomial p for a graph class such that every graph in the class has at most p( n) pieces then there is a polynomial time vertex ranking algorithm for this class. (Typically, a preprocessing determining auxiliary data is necessary to reach the best time bounds.)

The following classes of well· structured graphs have this property (see [17] for the polynomial number of minimal separators), which is always based on an intersection model (sometimes a 'circular' extension of a 'linear' model). Using this one can show that the vertex ranking can be done for interval graphs in O(n3), for trapezoid graphs in O(nS), for cocomparability graphs of dimension at most d in O( n3d ), for circular arc graphs in O( n3 ) and for circular permutation graphs in O( nS). (Details are omitted here.)

6 Open Problems

We like to mention the following open problems:

• It is not known whether the vertex ranking can be done in polynomial time for otber classes of well· structured graphs like chordal graphs and circle graphs .

• It is surprising that t.he algorithmic complexity of edge ranking on graphs in general is still open [25].

References

[1] B. Aspvall and P. Heggernes, Finding minimum height elimination trees for interval graphs in polynomial time, Technical Report No 80, Depart­ment of Informatics, University of Bergen, Norway, 1993.

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[2J C. Berge and C. Chvatal, Topics on Perfect Graphs, Annals of Discrete Math. 21, 1984.

[3J H. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Miiller and Z. Tuza, Rankings of graphs, manuscript.

[4J H.L. Bodlaender, J.R. Gilbert, H. Hafsteinsson and T. Kloks, Approx­imating treewidth, Pathwidth and minimum elimination tree height, Proc. 17th International Workshop on Graph-Theoretic Concepts in Computer Science WG'91, Springer-Verlag, Lecture Notes in Computer Science 570, (1982), pp. 1-12.

[5] H. Bodlaender, T. Kloks and D. Kratsch, Treewidth and pathwidth of permutation graphs, Proceedings of the 20th International Colloquium on Automata, Languages and Programming, Springer-Verlag, Lecture Notes in Computer Science 700, (1993), pp. 114-125.

[6J A. Brandstadt and D. Kratsch, On domination problems for permuta­tion and other graphs, Theor. Comput. Sci. 54, 181 - 198, 1987.

[7] A. Brandstadt, Special graph classes - a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU·199 (1991) Universitat Duisburg Gesamthochschule.

[8] J.S. Deogun and Y. Peng, Edge ranking of trees, Congressius Numer­antium 79, 19 - 28, 1990.

[9J I.S. Duff and J.K. Reid, The multifrontal solution of indefinite sparse symmetric linear equations, ACAI Transactions on Alathematical Soft­ware 9, (1983), pp. 302-325.

[10J M. Farber and M. Keil, Domination in permutation graphs, J. Algo­rithms 6, (1985), pp. 309-321.

[I1J T. GaJlai, Transitive orientierbaren Graphen, Acta Math. Sci. Hung. 18, (1967), pp. 25-66.

[12J M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Aca­demic Press, New York, 1980.

[13J A.V. Iyer, H.D. Ratliff and G. Vijayan, Parallel assembly of modular products-an analysis, PDRC, Technical Report 88-06, Georgia Institute of Technology, 1988.

[14J A.V. Iyer, H.D. Ratliff and G. Vijayan, On edge ranking problems of trees and graphs, Discrete Applied Mathef1wtics 30, (1991), pp. 43-52.

[15J M. Katchalski, W. McCuaig and S. Seager, Ordered colourings, Manuscript, University of Waterloo, 1988.

[16J T. Kloks, Treetvidth, Ph.D. Thesis, Utrecht University, The Nether· lands, 1993.

10

[17} T. Kloks, H. Bodlaender, H. Muller and D. Kratsch, Computing tree width and minimum fill· in: all you need are the minimal separators. To appear in Proceedings of the First Annual European Symposium on Algorithms (1993).

[18} C.E. Leisersoll, Area efficient graph layouts for VLSI, Pmc. 21st Ann. IEEE Symp. FOCS, (1980), pp. 270-281.

[19} J.W.H. Lill, The role of elimination trees in sparse factorization, SIAM Journal of Matrix Analysis and Applications 11, (1990), pp. 134-172.

[20} J. Nevins and D. Whitney, Editors, Concurrent Design of Products and Processes, McGraw· Hill, 1989.

[21} A.A. Schaffer, Optimal node ranking of trees in linear time, Information Pmcessing Letters 33, (1989/1990), pp. 91-96.

[22} P. Scheffler, Node ranking and Searching on Graphs (Abstract), in: U. Faigle and C. Hoede, Srd Twente Workshop on Graphs and Combi­natorial Optimization, Memorandum No.1132, (May 1993), Faculty of Applied Mathematics, University of Twente, The Netherlands.

[23} Arunabha Sen, Haiyong Deng and Sumanta Guha, On a graph parti­tion problem with application to VLSI layout, Information Pmcessing Letters 43, (1992), pp. 87-94.

[24} J. Spinrad, On comparability and permutation graphs, SIA M J. Compo 14, (1985), pp. 658-670.

[25} P. de la Torre R. Greenlaw and A. A. Schaffer, Optimal ranking of trees in polynomial time, Proc. 4th ACM Symp. on Discrete Algorithms, Austin, Texas (1993), pp. 138-144.

[26} Ming-Shing Yu, Lin Yu Tseng and Shoe-Jane Chang, Sequential and Parallel algorithms for the maximum-weight independent set problem on permutation graphs, Information processing Letters 46, (1993), pp. 7-11.

11

In this series appeared:

91/01 D. A1stein

91/02 R.P. NedeIllelt H.C.M. de Swart

91/03 J.P. Katoen L.A.M. Schoenmakers

91/04 E. v.d. Sluis A.F. v.d. Stappen

91/05 D. de Reus

91/06 K.M. van Hee

91/07 E.Poll

91/08 H. Schepers

91/09 W.M.P.v.d.Aalst

91/10 R.C.Backhouse P.J. de Bruin P. Hoogendijk G. Malcolm E. Voermans J. v.d. Woude

91/11 R.C. Backhouse P.J. de Bruin G.Malcolm E.Voermans J. van der Woude

91/12 E. van der Sluis

91/13 F. Rietman

91/14 P. Lemmens

91/15 A.T.M. Aerts K.M. van Hee

91/16 A.J.J .M. Marcelis

91/17 A.T.M. Aerts P.M.E. de Bra K.M. van Hee

Dynamic Reconfiguration in Distributed Hard Real-Time Systems, p. 14.

Implication. A survey of the different logical analyses "if...,then ... ", p. 26.

Parallel Programs for the Recognition of P-invariant Segments, p. 16.

Performance Analysis of VLSI Programs, p. 31.

An Implementation Model for GOOD, p. 18.

SPEC1FICATIEMETHODEN, een overzicht, p. 20.

CPO-models for second order lambda calculus with recursive types and sUbtyping, p. 49.

Terminology and Paradigms for Fault Tolerance, p. 25.

Interval Timed Petri Nets and their analysis, p.53.

POLYNOMIAL RELATORS, p. 52.

Relational CatamoIllhism, p. 31.

A parallel local search algorithm for the travelling salesman problem, p. 12.

A note on Extensionality, p. 21.

The PDB Hypermedia Package. Why and how it was built, p. 63.

Eldorado: Architecture of a Functional Database Management System, p. 19.

An example of proving attribute grammars correct: the representation of arithmetical expressions by DAGs, p.25.

Transforming Functional Database Schemes to Relational Represenlalions, p. 21.

91/18 Rik van Geldrop

91/19 Erik Poll

91/20 A.E. Eiben R.V. Schuwer

91/21 I. Coenen W.-P. de Roever I.Zwiers

91/22 G. Wolf

91/23 K.M. van Hee LJ. Somers M. Voorhoeve

91/24 A.T.M. Aerts D. de Reus

91/25 P. Zhou I. Hooman R. Kuiper

91/26 P. de Bra GJ. Houben I. Paredaens

91/27 F. de Boer C. Palamidessi

91/28 F. de Boer

91/29 H. Ten Eikelder R. van Geldrop

91/30 1.C.M. Baeten F.W. Vaandrager

91/31 H. ten Eikelder

91/32 P. Struik

91/33 W. v.d. Aalst

91/34 I. Coenen

91/35 F.S. de Boer I.W. Klop C. Palamidessi

Transformational Query Solving, p. 35.

Some categorical properties for a model for second order lambda calculus with subtyping, p. 21.

Knowledge Base Systems, a Formal Model, p. 21.

Asserlional Data Reification Proofs: Survey and Perspective, p. 18.

Schedule Management: an Object Oriented Approach, p. 26.

Z and high level Petri nets, p. 16.

Formal semantics for BRM with examples, p. 25.

A compositional proof system for real-time systems based on explicit clock temporal logic: soundness and complete ness, p. 52.

The GOOD based hypertext reference model, p. 12.

Embcdding as a tool for language comparison: On the CSP hierarchy, p. 17.

A compositional proof system for dynamic proces crcation, p. 24.

Correctncss of Acceptor Schemcs for Regular Languages, p. 31.

An Algebra for Process Creation, p. 29.

Some algorithms to decide the equivalence of recursive types, p. 26.

Techniques for designing efficient parallel programs, p. 14.

The modelling and analysis of queueing systems with QNM-ExSpect, p. 23.

Specifying fault tolerant programs in deontic logic, p. 15.

Asynchronous communication in process algebra, p. 20.

92/01 J. Cocnen J. Zwiers W.-P. de Roever

92/02 J. Coenen J. Hooman

92/03 J.CM. Baeten J.A. Bergstra

92/04 J.P.H. W.v.d.Eijnde

92/05 J.P.H.W.v.d.Eijnde

92/06 J.C.M. Baeten J.A. Bergstra

92/07 R.P. Nederpelt

92/08 R.P. NederpeJt F. Kamareddine

92/09 R.C. Backhouse

92/lO P.M.P. Rambags

92/11 R.C. Backhouse J.S.CP.v.d.Woude

92/12 F. Kamareddine

92/13 F. Kamareddine

92/14 J.CM. Baeten

92/15 F. Kamareddine

92/16 R.R. Seljce

92/17 W.M.P. van der Aalst

92/18 R.Nederpclt F. Kamareddine

92/19 J.CM.Baeten J.A.Bergstra S.A.Smolka

92/20 F.Kamareddine

92/21 F.Kamareddine

A noLe on compositional refinement. p. 27.

A compositional semantics for fault tolerant real-time systems, p. 18.

Real space process algebra, p. 42.

Program derivation in acyclic graphs and related problems, p. 90.

Conservative fixpoint functions on a graph, p. 25.

Discrete lime process algebra, pAS.

The fine-structure of lambda calculus, p. 1 lO.

On stepwise explicit substitution, p. 30.

Calculating the Warshall/Floyd path algorithm, p. 14.

Composition and decomposition in a CPN model, p. 55.

Demonic operators and monotype factors, p. 29.

Set theory and nominalisation, Part I, p.26.

Set theory and nominalisation, Part II, p.22.

The total order assumption, p. lO.

A system at the cross-roads of functional and logic programming, p.36.

Integrity checking in deductive databases; an exposition, p.32.

Interval timed coloured Petri nets and their analysis, p. 20.

A unified approach 10 Type Theory through a refined lambda-calculus, p. 30.

Axiomatizing Probabilistic Processes: ACP with Generative Probabilities, p. 36.

Arc Types for Natural Language? P. 32.

Non well-foundedness and type freeness can unify the interpretation of functional application, p. 16.

92/22 R. Nederpclt F.Kamareddine

92/23 F.Kamareddine E.Klein

92/24 M.Codish D.Dams Eyal Yardeni

92/25 E.Poll

92/26 T.H.W.Beelcn W.J.J.Stut PAC.Verkoulen

92/27 B. Watson O. Zwaan

93/01 R. van Oeldrop

93/02 T. Verhoeff

93/03 T. Verhoeff

93/04 E.H.L. Aarts J.H.M. Korst PJ. Zwietering

93/05 J.C.M. Baeten C. Verhoef

93/06 J.P. Veltkamp

93/07 P.O. Moerland

93/08 J. Verhoosei

93/09 K.M. van Hee

93/10 K.M. van Hee

93/11 K.M. van Hee

93/12 K.M. van Hce

93/13 K.M. van Hee

A useful lambda notation, p. 17.

Nominalization, Predication and Type Containment, p. 40.

Bottum-up Abstract Interpretation of Logic Programs, p. 33.

A Programming Logic for Fro, p. 15.

A modelling method using MOVIE and SimCon/ExSpect, p. 15.

A taxonomy of keyword pattern matching algorithms, p.50.

Deriving thc Aho-Corasick algorithms: a case study into the synergy of programming methods, p. 36.

A continuous version of the Prisoner's Dilemma, p. 17

Quicksort for linked lists, p. 8.

Deterministic and randomized local search, p. 78.

A congruence theorem for structured operational semantics with predicates, p. 18.

On the unavoidability of metastable behaviour, p. 29

Exercises in Multiprogramming, p. 97

A Formal Detcrministic Scheduling Model for Hard Real­Time Executions in DEDOS, p. 32.

Systems Engineering: a Formal Approach Part I: Systcm Conccpts, p. 72.

Systems Engineering: a Formal Approach Part II: Frameworks, p. 44.

Systems Engineering: a Formal Approach Part Ill: Modeling Methods, p. 101.

Systems Engineering: a Formal Approach Part IV: Analysis Methods, p. 63.

Systems Engineering: a Formal Approach Part V: Specification Language, p. 89.

92/22 R. Nederpelt F.Kamareddine

92/23 F .Kamareddine E.K1ein

92/24 M.Codish D.Dams Eyal Yardeni

92/25 E.Poll

92/26 T.H.W.Beelen W,J,J.Stut P.A.C.Verkoulen

92/27 B. Watson G. Zwaan

93/01 R. van Geldrop

93/02 T. Verhoeff

93/03 T. Verhoeff

93/04 E.H.L. Aarts J.H.M. Korst P,J. Zwietering

93/05 J.C.M. Baeten C. Verhoef

93/06 J.P. Veltkamp

93/07 P.D. Moerland

93/08 J. Verhoosel

93/09 K.M. van Hee

93/10 K.M. van Hee

93/11 K.M. van Hce

93/12 K.M. van Hcc

93/13 K.M. van Hee

93/14 J.C.M. Baeten J.A. Bergstra

A useful lambda notation, p. 17.

Nominalization, Predication and Type Containment, p. 40.

Bottum-up Abstract Interpretation of Logic Programs, p. 33.

A Programming Logic for Fro, p. IS.

A modelling method using MOVlE and SimCon/ExSpect, p. 15.

A taxonomy of keyword pattern matching algorithms, p.50.

Deriving the Aho-Corasick algorithms: a case study into the synergy of programming methods, p. 36.

A continuous version of the Prisoner's Dilemma, p. 17

Quickson for linked lists, p. 8.

Deterministic and randomized local search, p. 78.

A congruence theorem for structured operational semantics with predicates, p. 18.

On the unavoidability of metastable behaviour, p. 29

Exercises in Multiprogramming, p. 97

A Formal Deterministic Scheduling Model for Hard Real­Time Executions in DEDOS, p. 32.

Systems Engineering: a Formal Approach Part I: System Concepts, p. 72.

Systems Engineering: a Formal Approach Pan II: Frameworks, p. 44.

Systems Engineering: a Formal Approach Part III: Modcling Mcthods, p. 101.

Systems Engineering: a Formal Approach Part IV: Analysis Methods, p. 63.

Systems Engineering: a Formal Approach Pan V: Specification Language, p. 89.

On Sequential Composition, Action Prefixes and Process Prefix, p. 21.

93/15 J.c'M. Baelen I.A. Bergslra R.N. Bol

93/16 H. Schepers J. Hooman

93/17 D. Alstein P. van der Slok

93/18 C, Verhoef

93/19 G-J. Houbcn

93{2() F.S. de Boer

93121 M. Codish D. Dams G. File M. Bruynooghe

93/22 E. Poll

93/23 E. de Kogel

93{24 E. Poll and Paula Severi

93{25 H. Schepers and R. Gerth

93{26 W.M.P. van der Aalst

93127 T. Kloks and D. Kral,ch

93{28 F. Kamareddine and R. NederpclL

93{29 R. Post and P. De Bra

A Real-Time Process Logic, p. 31.

A Trace-Based Composilional Proof Theory for Faull Tolerant Distributed Syslems, p. 27

Hard Real-Time Reliable Multicast in the DEDOS system, p. 19.

A congruence theorem for structured operationaJ semantics with predicates and negative premises, p. 22.

The Desil,'Tl of an Online Help Facility for ExSpect, p.21.

A Process Algebra of Concurrent Constraint Program­ming, p. 15.

Freeness AnalySis for Logic Programs - And Correct­ness?, p. 24.

A Typechccker for Bijeclive Pure Type Systems, p. 28.

Relalional Algebra and Equalional Proofs, p. 23.

Pure Type Syslems with Definitions.

A Composilional Proof Theory for Fault Tolerant Real­Time Distribuled Syslems, p. 31.

MuW-dimensional Pelri nets, p. 25.

Finding all minimal separators of a graph, p. 11.

A Semantics for a fine A-calculus with de Bruijn indices, p.49.

GOLD, a Graph Oriented Language for Databases, p. 42.