Shigeru MASUYAMA + and Shin-ichi NAKAYAMA ++, Members SUMMARY This

INVITED SURVEY PAPER Special Issue on Algorithm Engineering:SurveysWhat Structural Features Make Graph Problems to Have Efficient Parallel Algorithms?—Using Outerplanar Graphs,Trapezoid Graphs and In-Tournament Graphsas Examples—Shigeru MASUYAMA†and Shin-ichi NAKAYAMA††,MembersSUMMARY This paper analyzes what structural features of graph problems allow efficient parallel algorithms.We survey some parallel algorithms for typical problems on three kinds of graphs,outerplanar graphs,trapezoid graphs and in-tournament graphs.Our results on the shortest path problem,the longest path problem and the maximumflow problem on outerplanar graphs,the minimum-weight connected dominating set problem and the coloring problem on trapezoid graphs and Hamiltonian path and Hamiltonian cycle problem on in-tournament graphs are adopted as working examples.key words:parallel graph algorithms,structure and complexity, outerplanar graph,trapezoid graph,in-tournament graph1.IntroductionDespite recent drastic increase of computing power available due to the development of computer technolo-gies,a number of computationally intensive applica-tions such as computer graphics,image processing,sim-ulations on the global environment,etc.,require much more computing power,which strongly motivates the research on parallel computation.A parallel algorithm is used in parallel computation executed on a paral-lel computer having multiple processors.It is known that an efficient parallel algorithm can be designed for a problem if it is decomposable into independently solv-able subproblems,i.e.,if divide-and-conquer is appli-cable.Moreover,some parallel algorithm design tech-niques have been devised like pointer jumping,prefix computation,balanced binary tree,Euler tour tech-nique and tree contraction,etc.(see e.g.,[14],[19], [36]).However,we still do not have enough knowledge to develop efficient parallel algorithms for a variety of problems encountered in the real world.Thus it is de-sired that structural features of problems which enable us to develop efficient parallel algorithms are clarified.As an attempt to contribute to this purpose,this Manuscript received July13,1999.Manuscript revised November24,1999.†The author is with the Department of Knowledge-Based Information Engineering,Toyohashi University of Technology,Toyohashi-shi,441–8580Japan.††The author is with the Department of Mathematical Sciences,Faculty of Integral Arts and Sciences,The Uni-versity of Tokushima,Tokushima-shi,770–8502Japan.paper analyzes what structural features make graph problems to have efficient parallel algorithms.We con-sider,as working examples,three kinds of graphs,out-erplanar graphs,trapezoid graphs and in-tournament graphs,which are known to allow efficient parallel al-gorithms for a number of graph problems.We sur-vey some parallel algorithms for typical problems on these three kinds of graphs including our results,e.g., the shortest path problem,the longest path problem and the maximumflow problem on outerplanar graphs, the minimum-weight connected dominating set prob-lem and the coloring problem on trapezoid graphs and Hamiltonian path and Hamiltonian cycle problem on in-tournament graphs.By doing this we try to elucidate structural features to allow efficient parallel algorithms so that useful insight to develop efficient parallel graph algorithms is obtained.PRAM(Parallel Random Access Machine,see e.g., [14],[19])is adopted as a parallel computation model throughout this paper,which is a kind of SIMD(Single Instruction Stream,Multiple Data Stream,see e.g., [14],[19])consisting of multiple processors with shared memory(see Fig.1).PRAMs are classified into the fol-lowing four classes with respect to access control on the shared memory:CRCW-PRAM,CREW-PRAM, ERCW-PRAMand EREW-PRAM,where C,E,R de-notes concurrent,exclusive,read and write,respec-tively(see e.g.,[14]).Theoretical computer scientistsFig.1PRAM.prefer PRAMfor design and analysis of parallel algo-rithms as,by this model,we can concentrate on the intrinsic parallelism in the problem under considera-tion without much worry about communication aspectamong processors which depends on the adopted par-allel architecture.In other words,we adopt divide-and-conquer strategy and concentrate,in this paper,on nonarchitectural aspects of the parallel computation.General background for developing parallel algo-rithms are found in e.g.,Refs.[14],[19],[36]and the reader may refer to e.g.,Refs.[3],[5],[18]on terminol-ogy on graph theory.2.Parallel Algorithms on Outerplanar Graphs The class of outerplanar graphs is a subclass of that of series-parallel graphs,a subclass of planar graphs. More efficient parallel algorithms than those for series-parallel graphs can often be obtained on outerplanar graphs by utilizing their structural features.An out-erplanar graph is an undirected graph where each bi-connected component can be embedded into the planeso that its peripheral(a cycle)has no diagonals which mutually intersect(see Fig.2).A graph has page num-ber one,i.e.,there exists a linear arrangement of ver-tices so that no pair of edges are crossing when they are drawn on the same side of the linear arrangementof vertices(see e.g.,[4],[7],[37]).Note that this prop-erty has application to VLSI design(see e.g.,[37]).The structural feature which allows outerplanar graphs to have efficient algorithms is the followingtree structure among faces,in addition to the trivial tree structure of bi-connected components,where bi-connected components sharing some vertex is consid-ered to be adjacent:For each bi-connected component G i,let T i=(V i,E i)where V i is the set of vertices each of which corresponds to a face and(u,v)in E i if and only if faces corresponding to u,v share someedge.Then T i is a tree(see e.g.,[14]).This makes divide-and-conquer applicable for developing parallelFig.2Outerplanar graph.algorithms,i.e.,first a local solution is constructed for each face and then a solution for T i is constructed in a bottom-up fashion from leaves to the root of T i by applying some standard parallel algorithm design tech-nique,e.g.,tree contraction.Finally,by also utilizing the aforementioned tree structure representing the rela-tion among bi-connected components,the solution for the given outerplanar graph is constructed in a manner similar to the case for each bi-connected component.Note that a number of efficient parallel algorithms customized for series-parallel graphs have a restriction so that two terminals,denoted by s,t,arefixed to those of series-parallel graphs.However,parallel algorithms may be devised on outerplanar graphs without such a restriction.The peripheral of a bi-connected component of an outerplanar graph can be obtained by using st number-ing[25],where st numbering is a one-to-one mapping from V to the set of integers{1,2,...,n}satisfying the following conditions[25](see Fig.3).(1)f(s)=1,f(t)=n.(2)For each vertex v∈V−{s,t},v is adjacent to some vertices v1,v2satisfying f(v1)<f(v)<f(v2).A graph G has an st numbering if and only if G is bi-connected[25].st numbering is a very useful tool for developing efficient parallel algorithms for a number of problems on bi-connected graphs,e.g.,centering prob-lem[31],which asks to construct a spanning tree T of a graph where the center,i.e.,the vertex from which the farthest vertex is the shortest,comes to the given vertex r.Note that st numbering can be obtained in O(log n)time using O(nα(m,n)/log n)processors on an ARBITRARY CRCW-PRAM[13].Hereα(m,n)de-notes the inverse of Ackermann function.This function increases very slowly with respect to m and n.It is obvious that a Hamiltonian cycle always exists in a bi-connected outerplanar graph as the peripheral of a bi-connected outerplanar graph is a Hamiltonian cycle.In addition to this,it is known that a bi-connected outer-planar graph having at least three vertices has a vertex with degree two[27].Consequently,for a bi-connected outerplanar graph,let a vertex with degree two be s and let a vertex adjacent to s be t,and execute st number-Fig.3st-numbering.MASUYAMA and NAKAYAMA:WHAT STRUCTURAL FEATURES MAKE GRAPH PROBLEMS543Fig.4A parallel algorithm for the shortest path on outerpla-nar graph.ing from1to n,we can prove that an edge{i,i+1} exists for each i,0≤i≤n−1,where vertex numbers are assigned by st numbering.Thus,if G is outerpla-nar,edge{i,i+1},i=1,···,n−1and{n,1},where vertex numbers are assigned by st numbering,exist and constructs Hamiltonian cycle C,which becomes the pe-ripheral.A parallel algorithm for obtaining the shortest path between any pair of vertices s,t in outerplanar graphs was proposed in[29]which runs in O(log n) time with O(n log log n/log n)processors on a CRCW PRAM.Outline of the algorithm is as follows(see Fig.4). Algorithm Shortest-Path1.Find the peripheral C of G by st numbering.2.Construct a circular list consisting ofvertexFig.5Rake operation.numbers1,2,...,n by generating a pointer for each edge{i,i+1},i=1,...,n−1and{n,1}constructing Hamiltonian cycle C in order to assign vertex number1 to vertex s.Vertices1,2,...,n are arranged clockwise on C and edges except those on C exist inside of C.3.Obtain array LEN(j),which shows the distance from s to j,to compute the distance between each pair of vertices on the peripheral by using prefix computa-tion on the list L i where the length of each edge{i−1,i} is stored.4.Construct directed cycles C1,...,C k by using numbers assigned to vertices as follows:Construct a di-rected graph G =(V ,E ),from G by orienting edges on C from i to i+1,i=1,...,n,where n+1is consid-ered to be1and adding two directed edges(j,k)and (k,j)for each diagonal{i,j}(see Fig.4(a)).C1,...,C k are directed cycles each of which corresponds to a face in C and each edge is an edge of G =(V ,E ).5.Construct tree T G whose vertices are directed cycles C1,...,C k of G,each of which corresponds to a face(see Fig.4(b)).6.Obtain the main line and branches of T G.Here main line is the set of edges on path c1,...,c from root c1of T G to c where c corresponds to a cycle C having an edge entering t in G .Other edges in T G are branches,and vertices on path c1,...,c are main vertices.7.Let T∗i,i=1,...,k−1,k be a subtrees corre-sponding to a branch obtained by removing main ver-tex and edges incident to them from T pute the shortest distance between vertices a i,b i shared by subgraph G∗i of G corresponding to T∗i and a cycle cor-responding to a vertex on the main line.Here G∗i is a subgraph of G consisting of an undirected cycle corre-sponding to T∗i.Tree contraction which uses the rake operation[19]illustrated in Fig.5is used for computing the shortest distance(see e.g.,Fig.6).pute LEN (j)for obtaining the distance between each pair of vertices on the peripheral of G∗where G∗consists of two paths P1and P2from s to t on the peripheral and diagonals connecting them(see Fig.4(d)).9.By combining shortest paths for G∗i’s,obtain the shortest path in G∗and we have the shortest path in G.✷Nakayama et al.[28]proposed a parallel algorithm544IEICE TRANS.INF.&SYST.,VOL.E83–D,NO.3MARCH2000Fig.6Computation of shortest distance using rake operationof tree contraction.for obtaining the longest path between arbitrary ver-tices s,t in an outerplanar graph G which runs inO(log n)time with O(n3/log2n+M(n))processors on a CRCW-PRAM.Throughout this paper,M(n)denotes the number of processors needed to compute the product of two n×n matrices in O(log n)time. Note that the currently-known upper bound is M(n)= O(n2.396)[21].Then,we can embed G in the plane so that the peripheral comes outside and diagonals con-nects some vertices on the peripheral without intersec-tion.Hence we have two vertex-disjoint paths connect-ing s,t.Then we construct acyclic digraph G∗having the longest path with the same length as that of G(see e.g.,Figs.7,8which illustrate how to construct G∗). Finally,we apply the parallel algorithm[21]for obtain-ing the longest path in acyclic digraphs.An O(log n)time parallel maximumflow al-gorithm with O(n2/log n)processors on a CREW-PRAMis proposed by Ref.[30].We briefly sketch the algorithm assuming that G(Fig.9)is bi-connected. It is straightforward to combine results for each bi-connected component to have the globalflow in G[30]. Algorithm Max-Flow1.Obtain the peripheral of G by st numbering ina manner similar to Algorithm Shortest-Path.Assign serial numbers1,2,...,n to vertices of G by starting from s.If G has no diagonal,then go to6.2.Assign the direction to each edge on the periph-eral C of G from vertex having smaller number to that Fig.7Outerplanar graph G and a cyclic digraph G∗corre-sponding to G.Fig.8The construction of acyclic digraph G∗. having the larger number.Replace each diagonal with two directed edges having converse directions.And we have a directed graph(a digraph,for short)G .G is an Euler graph as each vertex has the equal in-degree and out-degree.Construct cycles C1,...,C k of G accord-ing to vertex numbers in a manner similar to Algorithm Shortest-Path.MASUYAMA and NAKAYAMA:WHAT STRUCTURAL FEATURES MAKE GRAPH PROBLEMS545 Fig.9Outerplanar graph G.Fig.10T G whose vertices corresponds to C1,...,C k.Fig.11˜G.3.Construct tree T G whose vertices corresponds toC1,...,C k(Figure10illustrates T G constructed fromG in Fig.9).4.Obtain the main line of T G connecting the cyclecontaining s and the cycle containing t,and branchesobtained by deleting the main line in a manner similarto Algorithm Shortest-Path.Let subgraph of G consisting of the face corre-sponding to the main line of T G be˜G(see Fig.11).If there is no branch,then go to6.5.Let subtrees of T G corresponding to branchesbe T1,...,T k obtained by removing main vertices andedges incident to them.for all i,1≤i≤k in parallel doObtain the maximumflow from x i to y i which arevertices shared by the subgraph G i corresponding to T iand˜G.Minimum x i-y i cuts are obtained in parallel as fol-lows:Construct T∗i by appending the following verticesand edges:vertex v∗r,n vertices v∗1 ,...,v∗n where nFig.12The modified dual graph˜G∗.is the number of edges on the peripheral of G i.Edges(v∗i,r i)where r i is the root of T i and(v j,v k)wherev j in T i corresponds to face˜C j and v k corresponds toperipheral edge ekof˜C j.By Euler tour technique,the distance from v∗r toeach leaf of T∗i is obtained(see Fig.13(b)).6.Obtain the modified dual graph˜G∗of˜G(seeFig.12).(Note that the maximumflow of the pri-mal graph corresponds to the shortest path of the dualgraph.)Here˜G∗is obtained as follows:First constructdual graph˜G d of˜G.Then remove vertex v∗of˜G d cor-responding to the infinite face of G and add verticesv∗up,v∗down instead,and connect them to each vertexadjacent to v∗on upper(and lower)peripheral of G dwhich corresponds to the forward-side(and rear-side,respectively,)edge of˜G(,respectively).7.Let the length (i∗,j∗)of each edge(i∗,j∗)of˜G∗be the capacity c(i,j)of the edge(i,j)of˜G corre-sponding to (i∗,j∗),and obtain the shortest path fromv∗up to v∗down in˜G∗.✷Isomorphism of outerplanar graphs as well as treescan be tested in O(log n)time with n/log n proces-sors on a CRCW-PRAMand in O(log2n)time withn/log2n processors on an EREW-PRAM[22],This es-tablishes thefirst optimal parallel algorithm for graphisomorphism for a nontrivial class of graphs.A gen-eral optimal and very simple parallel algorithm of tree546IEICE TRANS.INF.&SYST.,VOL.E83–D,NO.3MARCH 2000compression which is useful for other problems is also proposed therein.Gibbons et al.[15]showed that minimum edge coloring on outerplanar graphs is in NC .Minimum edge coloring in outerplanar graphs is used in [6]as a subproblem for solving edge coloring on series-parallel graphs.Chrobak et al.[8]considered the problem ofori-Fig.13(a)Construction of T ∗i ,(b)directed Eulerian circuit obtained from T ∗i .Fig.14Trapezoid diagram.enting the edges of a graph in such a way that the out-degree of each vertex is minimized.An orientation is called d -bounded if for each vertex in the graph the out-degree is at most d .The subgraph isomorphism problem for two-connected outerplanar graphs is shown to be in NC 3,a subclass of NC ([24]).Whether a given graph is outerplanar or not can be decided in O (log n )time with O (nα( ,n )/log n )pro-cessors on a CRCW P RAM using st numbering [32].Here α( ,n )denotes the inverse of Ackermann function.This algorithm is near optimum in the sense that the product of time complexity and space complexity is al-most linear,i.e.,O (nα( ,n ))which outperforms that of [26].By combining algorithms in [10],[13]and [21]we can also decide if a given graph is outerplanar or not in parallel by the same time complexity.However,the algorithm in [32]is much more simpler.3.Parallel Algorithms on Trapezoid GraphsA trapezoid diagram consists of two horizontal lines called the top channel and the bottom channel,respec-tively,and n trapezoids where each trapezoid T has two corners having x coordinates a (T )and b (T ),re-spectively,on the top channel and two corners having x coordinates c (T )and d (T ),respectively,on the bot-tom channel,respectively(see Fig.14).The trapezoid graph G =(V,E )is a graph where there exists a trape-zoid diagram and each vertex corresponds one-to-one to a trapezoid in the diagram and there is an edge (v,w )if and only if trapezoids corresponding to v and w inter-sect in the diagram [9](see Fig.15).This class of graphs was first introduced by Dagan et al.[9].Trapezoid graphs are a generalization of permuta-tion graphs in that each line corresponding to a vertex of a permutation graph can be regarded as a trapezoid both of whose upper bottom and lower bottom have length 0.Moreover,trapezoid graphs can also be re-garded as a generalization of interval graphs in that an interval corresponding to a vertex can be regarded as a trapezoid of height 0where upper and lower bottoms coincide.Thus there is a possibility that some efficient algorithm may be obtained for a problem in trapezoidMASUYAMA and NAKAYAMA:WHAT STRUCTURAL FEATURES MAKE GRAPH PROBLEMS547Fig.15Trapezoid graph.graphs if it has some efficient algorithms in such graphs.One of the outstanding structural feature of atrapezoid graph is its co-comparability[9],where agraph G is co-comparable if and only if its comple-mentary graph G c has an antisymmetric and transitiveorientation F[16].This structural feature allows trape-zoid to have efficient parallel algorithms for a numberof graph problems.Note that minimum coloring of a trapezoid graph arises in a channel routing problem in VLSI design[12]. Given some labeled terminals on the upper and lower side of a two-sided channel,terminals with the same label will be connected in a common net.Such a net may be modeled by a trapezoid connecting the right-most resp.leftmost terminals with the same s then may be routed without intersection if and only if the corresponding trapezoids do not intersect,i.e.,if and only if they are independent.The number of col-ors needed to color the trapezoid graph is the number of layers needed to route the nets without intersection[12].In Ref.[34],we developed a parallel algorithm for solving minimum coloring which runs O(log2n)time with O(n3/log n)CREW-PRAMprocessors.This al-gorithm is based on the fact that minimum coloring on a graph G is equivalent to the decomposition prob-lem into cliques on the complementary graph of G.In particular,a clique corresponds to a path in the com-plementary graph with an orientation of a trapezoid graph by the aforementioned co-comparability.Conse-quently,minimum coloring is reduced to minimum path cover.In order to solve minimum path cover in the complementary graph with an orientation,let a point r i=(a(T i),c(T i))corresponding to a left-side edge of trapezoid T i be red,while let a point w i=(b(T i),d(T i)) corresponding to a right-side edge of trapezoid T i be white,and draw them in the plane.Then,trapezoids T i and T j do not intersect if and only if red point r j corresponding to trapezoid T i dominates white point w i corresponding to trapezoid T j(see Fig.16),where point p=(x,y)dominates point q=(z,u)if and only if z<x and y<u.Let r∈R can be matched with w∈W if and only if r dominates w and obtain the maximum matching, where R(W)is the set of red points(white points, respectively).Minimum coloring is obtained from the result.Nakayama et al.[33]proposed an O(log2n)time algorithm for solving minimum connected dominating Fig.16The geometric representation of the trapezoid graph in the plane.set using O(n3)processors on a CREW PRAMon weighted trapezoid graphs,i.e.,trapezoid graphs where each vertex has a weight.We utilize the fact that each path connecting an edge corresponding to the source pair and that corresponding to sink pair cor-responds to a connected dominating set,and we obtain the minimum connected dominating set by computing the shortest path among them.Parallel algorithms for minimum connected domi-nating set as well as independent dominating set,total dominating set on unweighted trapezoid graphs were proposed by Liang[23].They run in O(log2n)time on an EREW-PRAMwith O(max{n3/log2n,n2.376})pro-cessors for the independent domination problem,and O(max{nm2/log2n,m2.376})processors for the other domination problems,where m is the number of edges in a trapezoid graph of n vertices.4.Parallel Algorithms on In-TournamentA tournament graph(a tournament,for short)is a directed graph obtained by assigning orientation to each edge of a complete graph(see Fig.17).An in-tournament,a super class of a tournament,is a directed graph where if both(x,z)and(y,z)exists,then ex-actly one of(x,y)or(y,x)exists which always holds for tournaments.This structural feature makes some path problems including Hamiltonian path to have ef-ficient parallel algorithms.We developed,on an EREW-PRAM,an O(log2n) time algorithm using M(n)processors for deciding if a Hamiltonian path exists or not on in-tournament and construct it if it exists,and an O(log n)time Hamilto-nian cycle detection algorithm,which also construct it if it exits,using O(n+m)processors when a Hamilto-nian path is given as an input[35].Our algorithm is based on the following lemma.548IEICE TRANS.INF.&SYST.,VOL.E83–D,NO.3MARCH2000Fig.17In-tournament.Lemma1[2]:Connected in-tournament D has a Hamiltonian path if and only if D has an in-branching as a subgraph.Here in-branching is a directed graph where each vertex except root r has out-degree1,hav-ing no cycle,r can be reached by a directed path from any vertex.✷We only outline the aforementioned parallel Hamil-tonian path detection and construction algorithm as follows.Algorithm Hamiltonian-Path1.Construct an acyclic graph by making a vertex for each strongly connected component.2.Let a vertex with out-degree0be root r,re-verse the direction of each edge and execute breadth-first search to have an in-branching.3.Decompose the in-branching into subgraphs D1,D2,...,D s each of which consists of two merging paths.4.For each D i in parallel,obtain Hamiltonian pathH i.5.Merge H1,H2,...,H s to construct a Hamilto-nian path.✷Based on the Hamiltonian path x1,x2,···,x n con-structed by the above algorithm,a Hamiltonian cycle is constructed as follows:By replacing pair of backward edges(x j,x i),x i< x j with path x1,x2,···,x n,a path from x n to x1is constructed,andfinally a Hamiltonian path x1,···,x n,···,x1is constructed.5.ConclusionSome other special cases of graphs including chordal graphs(see e.g.,[20]),partial k-trees(see e.g.,[38]), e.t.c.,may allow a number of graph problems to have efficient parallel algorithms.In particular,it may be interesting to consider efficient parallel algorithms in k-trapezoid and circle trapezoid graphs[12],which are extensions of trapezoid graphs.Currently,design of parallel algorithms is still not at the stage of technology but that of art.Thus refining the theory so that parallel algorithm can be designed more easily and systematically remains for future re-search.AcknowledgementThe authors are grateful to anonymous referees whose comments were greatly beneficial to improving this pa-per.This work was partially supported by the Grant-in-Aid by the Ministry of Education,Science,Sports and Culture of Japan.References[1] E.Bampis,Y.Manoussakis,and is,“NC algorithmsfor antidirected Hamiltonian paths and cycles in tour-naments,”put.,vol.21, pp.223–234,1996.[2]J.Bang-Jensen,J.Huang,and E.Prisner,“In-tournamentdigraphs,”binatorial Theory,Series B.,vol.59, pp.267–287,1993.[3] C.Berge,Graphes et Hypergraphes,Dunod,1970.[4] F.Bernalt and P.C.Kainen,“The book thickness of agraph,”binatorial TheorySer B,vol.27,pp.320–331, 1979.[5]J.A.Bondy and Murty,Graph Theory with Applications,MacMillan,1976.[6]Y.Caspi and E.Dekel,“Edge coloring series parallelgraphs,”J.Algorithms18,no.2,pp.296–321,1995.[7] F.R.K.Chung,F.T.Leighton,and A.L.Rosenberg,“Em-bedding graphs in books:A graph layout problem with applications to VLSI design,”SIAM J.Algebraic Discrete Methods,vol.8,pp.33–58,1987.[8]M.Chrobak and D.Eppstein,“Planar orientations with lowout-degree and compaction of adjacency matrices,”put.Sci.86,no.2,243–266,1991.[9]I.Dagan,M.C.Golumbic,and R.Y.Pinter,“Trapezoidgraphs and their coloring,”Discrete Applied Mathematics, vol.21,pp.35–46,1988.[10]K.Dicks,T.Hagerup,and W.Wrytter,“Optimal paral-lel algorithms for the recognition and coloring outerplanar graphs,”LNCS379,Springer,1989.[11]K.S.Easwarakumar,C.P.Rangan,and G.A.Cheston,“Alinear algorithm for centering a spanning tree of a bi-connected graph,”Information Processing Letters,vol.51, pp.121–124,1994.[12]S.Felsner,R.Muller,and L.Wernisch,“Trapezoid graphsand generalizations,geometry,and algorithms,”Discrete Applied Mathematics,vol.74,pp.13–32,1997.[13] D.Fussel,V.Ramachandran,and R.Thurimala,“Findingtriconnencted components by local replacement,”SIAM J.Compt.,vol.22,pp.587–616,1993.[14] A.Gibbons and W.Rytter,Efficient Parallel Algorithms,Cambridge University Press,1988.[15] A.Gibbons and W.Rytter,“Optimally edge-colouring out-erplanar graphs is in NC,”put.Sci.,vol.71, no.3,pp.401–411,1990.[16]M.C.Golumbic,Algorithmic Graph Theory and PerfectGraphs,Academic Press,1980.[17] D.J.Haglin and M.J.Wolf,“On convex subsets in tour-naments,”SIAM J.Discrete Math.,vol.9,no.1,pp.63–70, 1996.[18] F.Harary,Graph Theory,Addison-Wesley,1968.[19]J.J´a J´a,An Introduction to Parallel Algorithms,Addison-Wesley,1992.[20]P.N.Klein,“Efficient parallel algorithms for chordalMASUYAMA and NAKAYAMA:WHAT STRUCTURAL FEATURES MAKE GRAPH PROBLEMS549graphs,”SIAM put.vol.25,no.4,pp.797–827,1996.[21]Jan van Leewen,ed.,Handbook of Theoretical ComputerScience vol.A:Algorithms and Complexity,Elsevier,1990.[22] C.Levcopoulos, A.Lingas,O.Petersson,and W.Ryt-ter,“Optimal parallel algorithms for testing isomorphism of trees and outerplanar graphs,”Foundations of Software Technology and Theoretical Computer Science,pp.204–214, Lecture Notes in Comput.Sci.,vol.472,Springer,Berlin, 1990.[23]Y.D.Liang,“Parallel algorithms for the domination prob-lems in trapezoid graphs,”Discrete Appl.Math.vol.74, no.3,pp.241–249,1997.[24] A.Lingas and A.Proskurowski,“On parallel complexityof the subgraph homeomorphism and the subgraph isomor-phism problem for classes of planar graphs,”-put.Sci.,vol.68,no.2,pp.155–173,1989.[25]Y.Maon,B.Schieber,and U.Vishkin,“Parallel ear decom-position search(EDS)and st-numbering ib graphs,”Theo-retical Computer Science,vol.47,pp.277–298,1986. [26]S.Masuyama and S.Naito,“Deciding whether graph Ghas page number one is in NC,”Information Processing Letters,vol.44,pp.7–10,1992.[27]S.L.Mitchell,“Linear algorithms to recognize outerplanarand maximal outerplanar graphs,”Inf.Processing Lett., vol.9,pp.229–232,1979.[28]S.Nakayama and S.Masuyama,“A parallel algorithm forsolving the longest path problem in outerplanar graphs,”IEICE Trans.,vol.J78-D-I,no.6,pp.563–568,June1995. [29]S.Nakayama and S.Masuyama,“A parallel algorithm forfinding an st-shortest path in outerplanar graphs,”IEICE Trans.,vol.J78-D-I,no.11,pp.867–877,Nov.1995.[30]S.Nakayama and S.Masuyama,“A parallel algorithm forfinding the maximumflow in outerplanar graphs,”IEICE Trans.,vol.J79-D-I,no.5,pp.226–236,May1996.[31]S.Nakayama and S.Masuyama,“A parallel algorithm forcentering a spanning tree of a biconnencted graph,”IEICE Trans.,vol.J79-D-I,no.5,pp.299–302,May1996.[32]S.Nakayama and S.Masuyama,“A simple near optimalparallel algorithm for recognizing outerplanar graphs,”J.Math.Tokushima University,vol.30,pp.71–80,1996. [33]S.Nakayama and S.Masuyama,“A parallel algorithm forfinding a minimum-weight connected dominating set on trapezoid graphs,”Mathematica Japonica,vol.45,no.1, pp.165–171,1997.[34]S.Nakayama and S.Masuyama,“A parallel algorithm forsolving the coloring problem on trapezoid graphs,”Infor-mation Processing Letters62,pp.323–427,1997.[35]S.Nakayama and S.Masuyama,“Parallel algorithms forfinding a Hamiltonian path and a Hamiltonian cycle in an in-tournament graph,”IEICE Trans.Fundamentals, vol.E81-A,no.5,pp.757–767,May1998.[36]J.H.Reif,ed.,Synthesis of Parallel Algorithms,MorganKaufmann,1993.[37]M.Yanakakis,“Embedding planar graphs in four pages,”put.System Sci.,vol.38,pp.36–67,1989.[38]X.Zhou,S.Nakano,and T.Nishizeki,“Edge-coloring Par-tial k-trees,”J.Algorithms,vol.21,pp.598–617,1996.Shigeru Masuyama is a Professorat the Department of Knowledge-BasedInformation Engineering,Toyohashi Uni-versity of Technology.He received theB.E.,M.E.and D.E.degrees in Engineer-ing(Applied Mathematics and Physics)from Kyoto University,in1977,1979and1983,respectively.He was with the De-partment of Applied Mathematics andPhysics,Faculty of Engineering,KyotoUniversity from1984to1989.He joined the Department of Knowledge-Based Information Engineering, Toyohashi University of Technology in1989.His research in-terest includes computational graph theory,combinatorial opti-mization,parallel algorithms and natural language processing. Dr.Masuyama is a member of the OR society of Japan,the In-formation Processing Society of Japan,the Institute of Systems, Control,Information Engineers of Japan and the Association for Natural Language Processing ofJapan.Shin-ichi Nakayama is a Lecturerat the Department of Mathematical Sci-ences,the University of Tokushima.Hereceived B.E.,M.E.and Dr.of Engineer-ing from Toyohashi University of Technol-ogy,in1991,1993and1996,respectively.His research interests include graph the-ory,network theory and parallel algo-rithms.。