On the maximum connected interval subgraph of block grraphs and chain graphs

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(1)ON THE MAXIMUM CONNECTED INTERVAL SUBGRAPH OF BLOCK GRAPHS AND CHAIN GRAPHS , Hsiao-An Chao, Ruay-Shiung Chang. Sheng-Lung Peng. Department of Computer Science and information engineering, National Dong Hwa University, Hualien, Taiwan, R.O.C. Email:flung, m8721011, rschangg@csie.ndhu.edu.tw. Abstract. ments [2,3,10,8,15]. In this paper, we deal with the problem of

(2) nding an interval subgraph of the input graph with the maximum size. The maximum interval subgraph problem (M I SP for short) on G is the problem of

(3) nding a maximum interval subgraph of G by deleting the minimum number of vertices of G. A more generalized problem is referred as the node deletion problem [11,12,17]. Similarly, the maximum connected interval subgraph problem (M CI SP for short) on G is the problem of

(4) nding a maximum connected interval subgraph of G by deleting the minimum number of vertices of G. Note that the solution of MISP may not contain the solution of MCISP for a graph. For example, please see Figure 2.. The maximum connected interval subgraph problem on a graph G is the problem of

(5) nding an induced subgraph H of G such that H is a connected interval graph with the maximum number of vertices. It has been shown that this problem is NP-hard and have no constant ratio approximation on general graphs. In this paper, we propose linear-time algorithms for solving this problem on block graphs and chain graphs.. 1. Introduction Let G be a

(6) nite and simple graph with vertex set V (G) and edge set E (G). A graph G is an interval graph if there exists a one-to-one correspondence between V (G) and a family F of intervals of the real line such that two vertices in V (G) are adjacent if and only if their corresponding intervals in F overlap. Interval graphs have many applications, among them scheduling, seriation in archeology, medical diagnosis, behavioral psychology, circuit design and most recently the Human Genome Project [4{6,9,14].. a. b. c e G. d. b. (a). (c). Figure 2. (a) A graph G. (b) A solution of MISP. (c) A solution of MCISP. It has been shown that MCISP is NP-hard on general graphs [16]. Furthermore, it has also been shown that it is impossible to be approximated with ratio n1 , for every > 0, in polynomial time unless P = N P , where n is the number of vertices in the input graph [13]. So far, to our knowledge, there is no result for MCISP on any special graphs.. d. a. (b). c e F. In this paper, we solve this problem on block graphs and chain graphs. On a block graph, we show that this problem can be reduced to the problem of

(7) nding a maximum caterpillar (with a constraint) in its block tree. On a chain graph, we show that this problem is equivalent to the problem of

(8) nding a maximum caterpillar of this graph. As a result, we propose linear-time algorithms for solving this problem on above graphs.. Figure 1. An interval graph and its corresponding interval family. Recently, many researchers focus on

(9) nding an interval supergraph of the input graph with di erent require corresponding author: lung@csie.ndhu.edu.tw 1.

(10) The rest of this paper is organized as follows. In Section 2, we give the de

(11) nitions and notation used in this paper. The main results on block graphs and chain graphs are presented in Sections 3 and 4, respectively. Finally, we give the concluding remarks in the last section.. 12. 16. 9. 15. 1 2. 3. 4. 5. 8. 6. 7. 11. 17. 14. 10. 13. 2. De

(12) nitions and Notation Figure 3. A block graph G with 17 vertices.. Let G be a connected, simple and

(13) nite graph. Let V (G) and E (G) be its vertex and edge sets, respectively. Let n = jV (G)j and m = jE (G)j. Let N (v ) = fw j (v; w) 2 V (E )g denote the neighborhood of vertex v . For a vertex set W V (G), let G[W ] denote the subgraph of G induced by W . For two vertex sets X and Y , X nY = fv 2 X j v 2= Y g. In the following, \subgraph" means \induced subgraph.". 3. The block tree of a block graph can be constructed in linear time by a depth

(14) rst search [1]. For a block vertex u in a block tree, let Bu denote its corresponding block in G. Note that every leaf of a block tree is a block vertex.. A clique in a graph G is a complete subgraph of G. A clique also refers to a set of vertices whose induced subgraph is complete when there is no confusion in the description. An independent set I in G is a vertex subset of V (G) in which no two vertices are adjacent. For a graph G, a sequence of vertices hv1 ; v2 ; :::; vr i is a path if (vi ; vi+1 ) 2 E (G), 1 i r 1.. 12. 16. B5 1. 9. 2. 15. 3 9. 4. 5. 8 5. 6. 8. 5. 15. 11. B4. B9. B1. 17. 11 11. 10. B3. B2. For a tree T , a vertex with degree 1 is called a leaf and a non-leaf vertex is called an internal vertex. A tree T is called a caterpillar if V (T ) can be partitioned into two sets B and H such that vertices in B induce a path and vertices in H induce an independent set. The path is also called the backbone of this caterpillar. We call a vertex in B (respectively, H ) a backbone vertex (respectively, hair vertex).. 15. B7. B1. 5. B3. B2. 7. 8. 9. B7. B4. 11. 14. 14. B6. 7. B8. B5. B8. 13. (a). B6. (b). Figure 4. (a) The 9 blocks of G where a dark vertex denotes a cut vertex. (b) The block tree of G.. For a block graph G, we assume that TG is a caterpillar. By the de

(15) nition, TG can be partitioned into two sets B and H . Without loss of generality, let B = hv10 ; v20 ; : : : ; vp0 i be the backbone of TG , where v10 ; v30 ; : : : ; vp0 are block vertices and v20 ; v40 ; : : : ; v(p 1)0 are cut vertices. Note that every vertex in H must be a block vertex. Therefore, it is impossible to have a hair vertex who is adjacent to a block vertex in B since TG is a caterpillar. Let di be the number of the neighbors of vi0 in H for i 2 f1; : : : ; pg. Let vi1 ; vi2 ; : : : ; vidi be the block vertices which are the neighbors of vi0 in H . For a vertex u in G, we construct an interval Iu as follows. If u is not a cut vertex and belongs to block Bvij , let Iu = (lij ; lij + 0:5) where. 3. Block Graphs In this section, we discuss MCISP on block graphs. For any graph G, a vertex v is called a cut vertex if deleting v increases the number of connected components. A block is a maximal connected subgraph of G without any cut vertex. A graph is called a block graph if and only if its blocks are complete graphs and the intersection of two blocks is either empty or a cut vertex [7]. Note that trees form a subclass of block graphs. In the following, G is treated as a connected block graph. Suppose G has m blocks B1 ; B2 ; : : : ; Bm and n cut vertices v1 ; v2 ; : : : ; vn . The block cut vertex structure TG is the tree with vertex set V (TG ) = fB1 ; B2 ; : : : ; Bm ; v1 ; v2 ; : : : ; vn g and edge set E (TG ) = f(Bi ; vj ) j 1 i m; 1 j n; vj 2 Bi g. Vertices Bi , 1 i m, are called block vertices and vertices vj , 1 j n, are called cut vertices. For convenience, TG is also called the block tree of G. For example, Figure 4 shows the block tree of the graph G depicted in Figure. lij. 8 < = l : l (i. 1. +1 1) + 1. 1)d. i(j. i. 1. if i = 1 if i = 1; j = if i = 1; j >. 6. 6. 0 0. Otherwise, u is a cut vertex. In this case, assume that u = vi0 for some i. Then Iu = (l(i 1)0 ; l(i+1)0 + 0:5). 2. B9.

(16) Let F = fIu j u 2 Gg. It is not hard to see that there is a one-to-one correspondence between V (G) and F such that u and w are adjacent if and only if their corresponding intervals Iu and Iw intersect. That is, G is a connected interval graph.. Algorithm: MCISB; Input: A rooted block tree with root ; Output: The number of vertices of the maximum connected TG. u. interval subgraph of G; 1. initially, every vertex v in TG , label(v) = (0; 0); 2. label(u) = compute block tree label(u); 3.

(17) nd a vertex v such that either type1(v) or type2(v) is maximum; 4. Output= maxftype1(v); type2(v)g;. On the other hand, if TG is not a caterpillar, then G is impossible to be an interval graph. Hence, we have the following theorem.. Function: (vertex: ) 1. if is a leaf then ( ) = (j j 0); 2. else is not a leaf 3. for all vertices 1 , the children of , do ( ); 4. ( )= 5. next 6. if is a block vertex then compute block tree label. Theorem 3.1. u. A block graph G is a connected interval. graph if and only if. TG. label u. =. is a caterpillar.. u. =. vi ;. i. d. d. u. compute block tree label vi. label vi. According to Theorem 3.1, MCISP on a block graph G is equivalent to the problem of

(18) nding the maximum subgraph G0 of G such that TG0 is a caterpillar. In other words, our problem can be reduced to the problem of

(19) nding a maximum caterpillar in TG such that every hair vertex is a block vertex.. u. Bu ;. i. 7.. 8. 9. 10. 11. 12.. In the following, we treat TG as a rooted tree. A maximum caterpillar of a rooted block tree is also a maximum caterpillar of its underlying unrooted block tree. For a rooted tree T and a vertex u, let T [u] denote the subtree of T rooted at u. For convenience, we use G[TG [u]] to denote the block graph with block tree TG [u]. Therefore, if C is a caterpillar of TG , then G[C ] is the block graph with C being its block tree. For a rooted caterpillar, we de

(20) ne the following two types on its backbone vertices u.. 13. 14. 15. 16. 17. 18. 19.. u. let (a1 ; b1 ); (a2 ; b2 ); :::; (ad ; bd ) be the labels of , with a1 a2 ::: ad ; type1(u) = a1 1 + jBu j; if d = 1 then type2(u) = 0 else type2(u) = a1 + a2 2 + jBu j; else = u is a cut vertex = let (a1 ; b1 ); (a2 ; b2 ); :::; (ad ; bd ) be the labels of v1 ; v2 ; :::; vd , with a1 jBv1 j a2 jBv2 j ::: ad jBvd j; type1(u) = a1 + (jBvi j 1); i2f2;3;:::;dg if d = 1 then type2(u) = 0 else type2(u) = a1 + a2 1 + ( j Bv j 1); v 2 N (u)nfv1 ;v2 g v1 ; v2 ; :::; vd. P. P. end if ( )=( end if return ( ); label u. 1(u); type2(u));. type. label u. Type I:. Figure 5. Algorithm MCISB.. Type II:. Lemma 3.1. u has at most one child who is also a backbone vertex. u has exactly two children who are also backbone vertices.. If u is an internal block vertex of a block TG , (a1 ; b1 ); (a2 ; b2 ); : : : ; (ad ; bd ) are the labels of v1 ; v2 ; : : : ; vd , the d children of u, where a1 a2 ::: ad , then type1(u) = a1 1 + Bu . tree. j j Proof. By the de

(21) nition, type1(u) = jV (G[C ])j where . Note that for a rooted caterpillar, there is at most one backbone vertex which is Type II.. is the maximal caterpillar in TG [u] including u as a Type I vertex. If u is a block vertex, then V (G[C ]) consists of two sets, V (G[Ci ]) and V (Bu ) for some i, where G[Ci ] is the maximum caterpillar, in TG [vi ] including vi as a Type I vertex. Therefore, C. For a vertex u in TG , let type1(u) (respectively, type2(u)) denote jV (G[C ])j if there is a maximal caterpillar C in TG [u] with u being a Type I (respectively, Type II) vertex. In the case that u cannot be a Type II vertex, we let type2(u) = 0. Algorithm MCISB (Figure 5)

(22) rst computes (type1(u),type2(u)) for every vertex u in TG from leaves to the root and then

(23) nds the maxv2V (TG ) ftype1(v ); type2(v )g which is the number of vertices of the maximum connected interval subgraph of G.. type1(u). By the de

(24) nitions, for a leaf u in TG , type1(u) = jBu j and type2(u) = 0. Furthermore, for an internal vertex u, if u has just one child, then type2(u) = 0. Otherwise, we have the following lemmas for an internal block vertex.. = maxfjV (G[Ci ]) [ V (Bu )jg = maxfjV (G[Ci ])j 1 + jV (Bu )jg = maxfai 1g + jBu j = a1 1 + jBu j. ut Lemma 3.2 tree. 3. TG ,. If. u. is an internal block vertex of a block. (a1 ; b1 ); (a2 ; b2 ); : : : ; (ad ; bd ). are the labels of.

(25) v1 ; v2 ; : : : ; vd , the d children of u, ::: ad , d > 1, then type2(u) = a1. . where. a1. . a2. . Proof. The correctness of Algorithm MCISB is di-. + a2 2 + jBu j. Proof. By the de

(26) nition, type2(u) = jV (G[C ])j where C is the maximum caterpillar in TG [u] which includes u as a Type II vertex. Because there are more than one child, the backbone of C must have a subpath hvi ; u; vj i, where vi and vj are two children of u. If u is a block vertex, V (G[C ]) consists of three sets, V (G[Ci ]), V (G[Cj ]) and V (Bu ) for some i and j , where Ci (respectively, Cj ) is the maximum caterpillar in TG [vi ] (respectively, TG [vj ] including vi (respectively, vj ) as a Type I vertex. Therefore, type2(u). rectly from Lemmas 3.1, 3.2, 3.3 and 3.4. It is not hard to see that the Step 1 of MCISB takes O(n) time. Since the label of each vertex u in TG can be computed according to the labels of its children, it takes O(deg(u)) time. Therefore, totally, it takes O(n) time to compute all the labels. That is, the Step 2 of MCISB takes O(n) time. By using a standard search algorithm, Step 3 of MCISB can be done in O(n) time. Hence, we have that the time complexity of MCISB is O(n). ut Once the labels of vertices in TG are computed, we can use them to identify the maximum caterpillar C . First, we

(27) nd the vertex v with the maximum in ftype1(v); type2(v)g. Then, by a backtracking traversal on TG [v ], we can identify C . Finally, G[C ] can be found. This procedure also runs in linear time. Conclusively, we have the following theorem.. = maxfjV (G[Ci ]) [ V (G[Cj ]) [ V (Bu )jg = maxfai 1 + aj 1g + jBu j = a1 + a2 2 + jBu j. ut. Theorem 3.2. Using a similar argument, we have the following lemmas for a cut vertex.. Lemma 3.3. If. j. j j j j P j j. j. If u is a cut vertex of a block tree TG , (a1 ; b1 ); (a2 ; b2 ); : : : ; (ad ; bd ) are the labels of v1 ; v2 ; : : : ; vd , the d children of u, where a1 Bv1 a2 Bv2 ad Bvd , d > 1, then type2(u) = a1 + a2 1 + v2N (u)nfv1 ;v2 g ( Bv 1).. j. j. j. connected. interval. sub-. In this section, we consider MCISP on chain graphs. Let G = (X; Y; E ) be a connected bipartite graph with X = fx1 ; x2 ; : : : ; xp g and Y = fy1 ; y2 ; : : : ; yq g. The graph G is called a chain graph if and only if the neighborhoods of the vertices of X form a chain, i.e., the neighborhoods of the vertices of X can be ordered such that N (x1 ) N (x2 ) N (xp ). It is not hard to see that the neighborhoods of vertices in Y also forms a chain (N (y1 ) N (y2 ) N (yq )) [17]. For example, see Figure 7. By de

(28) nition, N (x1 ) = Y and N (y q ) = X .. Lemma 3.4. j P . maximum. 4. Chain Graphs. u is a cut vertex of a block tree TG , (a1 ; b1 ); (a2 ; b2 ); : : : ; (ad ; bd ) are the labels of v1 ; v2 ; : : : ; vd , the d children of u, where a1 Bv1 a2 Bv2 ad Bvd , d > 1, then type1(u) = a1 + 1) u2fv2 ;v3 ;:::;vd g ( Bu. j. A. graph of a block graph can be computed in linear time.. j. j j. An example for Algorithm MCISB is presented in Figure 6.. x1. x2. x3. x4. y2. y3. y4. y5. (11,0) (10,0)B5 9 (10,15) (7,0)B4 (6,9) (7,8) 8 (3,0)11 (4,0) (5,0) B3 (2,0) B6 (5,0) 5 7 (2,0). B1. B4 8. B3. B 7 (3,4) (2,0) 14. B8. B2. (2,0). B9. 11. B6. y1. B7. 5. 14. B1. B8. (a). B9. Figure 7. A chain graph.. (b). Figure 6. (a) An example of the labeling on a block tree. (b) The maximum caterpillar.. If a chain graph has at most three vertices, the maximum connected interval subgraph is this chain graph. In the following, we only consider the connected chain graphs with more than three vertices.. Lemma 3.5. Theorem 4.1. For a block graph. G,. Algorithm MCISB. correctly computes the number of vertices of the maximum connected interval subgraph of. G. A chain graph. graph if and only if. in linear time.. internal vertices.. 4. G. G. is a connected interval. is a caterpillar with at most two.

(29) Proof. First, it is not hard to see that if. G is not a caterpillar, then G is not an interval graph. Therefore, we consider that G is a caterpillar. If G = (X; Y; E ) has at most two internal vertices, it is also not hard to check that G is an interval graph. If G contains more than two internal vertices, then at least two vertices appear in one partite. Without loss of generality, we assume that xi and xj are internal vertices in X and N (xi ) N (xj ) and fyk ; yl g N (xj ). Then the vertices fxi ; yk ; xj ; yl g forms a 4-cycle. This contradicts that G is a caterpillar. Similarly, if at least two internal vertices appear in Y , then we also get a contradiction. Hence, this lemma holds. ut. y1. X. =. For a chain graph. fx ; x ; : : : ; xp g 1. 2. and. Y. =. G. = (X; Y; E ). fy ; y ; : : : ; yq g 1. 2. G. Y. with. , there. (x1 ; yq ). Proof. Let G1 = (X1 ; Y1 ; E1 ) be a maximum connected interval subgraph of G. By Theorem 4.1, G1 is a caterpillar with at most two internal vertices. We have the following cases.. X. x1 ; x2 ; : : : ; xp. g. =. fx ; x ; : : : ; xp g 1. 2. and. Y. =. fy ; y ; : : : ; yq g 1. 2. , the. number of vertices in a maximum connected interval. min2ip+1 fi + jN (xi )jg N (xp+1 ) = fyq g. Proof. Let G = (X ; Y ; E ) be a maximum connected interval subgraph of G satisfying Lemma 4.1. That is, (x1 ; yq ) 2 E . Let X = fx1 ; x1 ; x2 ; : : : ; xr g where 1 < 1 < < r p. Let X^ = X nfx1 g and Y^ = nfyq g. Because G is an interval graph, there is no Y edge between X^ and Y^ . Therefore, N (xi ) \ Y^ = ;, for i 2 f1; : : : ; rg. Since N (x1 ) N (x2 ) ^ = ;. We have N (xr ), we only consider N (x1 ) \ Y the following equations: subgraph of where. Case 2: G1 has one internal vertex. Without loss of generality, suppose that xi is the internal vertex for some i. Then X1 = fxi g and Y1 = Y . Let G2 = (fx1 g; Y; f(x1 ; y ) j y 2 Y g). Since N (xi ) N (x1 ), jV (G2 )j jV (G1 )j. That is, G2 is also a solution of MCISP which contains the edge (x1 ; yq ).. 1. xp+1. G. is. p+q. +3. is a dummy vertex and. n . ) jY^ j q jN (x1 )j ) +r 1p jV (G )j = jX^ [ Y^ [ fx ; yq gj = jX^ j + jY^ j + 2 r + q jN (x1 )j + 2 p + 1 + q jN (x1 )j + 2 p + q + 3 jN (x1 )j. ^ (Y N (x1 )) Y < 1 < < r p. 1. 1. Note that it is impossible for G1 to have no internal vertex since jV (G)j = p + q > 3. Hence, this lemma holds. ut. 1. 1. A solution of the graph depicted in Figure 7 is presented in Figure 8. By using Lemma 4.1, we have an algorithm to solve the MCISP on chain graphs (Figure 9). = (X; Y; E ). X. Figure 9. Algorithm MCISC.. Case 1: G1 has two internal vertices. Without loss of generality, let these two vertices be xi and yj for some i and j . If i 6= 1, then let G2 = (X2 ; Y2 ; E2 ) where X2 = (X1 nfxi g) [ fx1 g, Y2 = Y1 and E2 = f(x; y) j x 2 X2 ; y 2 Y2 ; (x; y) 2 E g. It is not hard to check that G2 is a caterpillar and jV (G1 )j = jV (G2 )j. Therefore, G2 is also a solution of MCISP. Similarly, if j 6= q , then we can obtain a graph G3 = (X3 ; Y3 ; E3 ) from G2 with X3 = X2 , Y3 = (Y2 nfyj g) [ fyq g and E3 = f(x; y ) j x 2 X3 ; y 2 Y3 ; (x; y ) 2 E g. It can be checked that G3 is a caterpillar and jV (G3 )j = jV (G2 )j. That is, G3 is also a solution of MCISP with desired condition.. G. X; Y ; E. y1 ; : : : ; yq. G; 1. add a dummy vertex xp+1 in X and let N (xp+1 ) = fyq g; 2. for i = 2 to p + 1 3. wi = (i 2) + (jN (xi )j 1); 4. next i 5. let wk = min2ip+1 fwi g; 6. if k > 2 then S = fx2 ; :::; xk 1 g [ (N (xk )nfyq g) 7. else S = N (x2 )nfyq g; 8. Output G[V (G) n S ];. contains the edge. For a chain graph. y5. y3. Algorithm: MCISC; Input: a chain graph = ( ) with = f and = f g; Output: a maximum connected interval subgraph of. exists a maximum connected interval subgraph which. Lemma 4.2. y2. x4. Figure 8. A maximum connected interval subgraph of the chain graph depicted in Figure 7.. According to Theorem 4.1, our problem on a chain graph G becomes to the problem of

(30) nding the maximum caterpillar of G with at most two internal vertices.. Lemma 4.1. x3. x1. It is not hard to see that the maximum occurs in 1 + jN (x1 )j = min2ip+1 fi + jN (xi ))jg. This completes our proof. ut The correctness of Algorithm MCISC is due to Lemma. with. 5.

(31) 4.2. It is not hard to see that the time complexity of MCISC is linear. Conclusively, we have the following theorem.. Theorem 4.2. A. maximum. connected. interval. 9. R.M. Karp. Mapping the genome: some combinatorial problems arising in molecular biology. 25th Ann. Sympos. on Theory and Computing, pages 278{285, 1993. 10. L.M. Kirousis and C.H. Papadimitriou. interval graph and searching. Discrete Mathematics, 55:181{184, 1985. 11. M.S. Krishnamoorthy and N. Deo. Node-deletion NP-complete problems. SIAM Journal on Computing, 8:619{625, 1979. 12. J.M. Lewis and M. Yannakakis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20:219{ 230, 1980. 13. C. Lund and M. Yannakakis. The approximation of maximum subgraph problems. Proc. 20th Inter-. sub-. graph of a chain graph can be computed in linear time.. 5. Concluding Remarks In this paper, we have proposed linear-time algorithms for the maximum connected interval subgraph problem on block graphs and chain graphs. It has been shown that both the maximum interval subgraph problem and the maximum connected interval subgraph problem are NP-hard and have no constant ratio approximation. For the maximum interval subgraph problem, Yannakakis showed that it remains NP-hard on bipartite graphs [17]. Therefore, it is interesting to decide whether the maximum connected interval subgraph problem on bipartite graphs is NP-hard or not. Besides, it is also interesting to study the maximum connected interval subgraph problem on other special graphs such as permutation graphs and chordal graphs (or even the split graphs). On the other hand, it is still unknown that the maximum interval subgraph problem is solvable on block graphs and chain graphs.. national Colloquium on Automata, Languages and. Programming, pages 40{51, 1993. 14. I. Pe'er and R. Shamir. Realizing interval graphs with side and distance constraints. SIAM Journal on Discrete Mathematics, 10:662{687, 1997. 15. S.L. Peng, M.T. Ko, C.W. Ho, T. s. Hsu, and C.Y. Tang. Graph searching on some subclasses of chordal graphs. Algorithmica, 27:395{426, 2000. 16. M. Yannakakis. The e ect of a connectivity requirement on the complexity of maximum subgraph problems. Journal of the Association for Computing Machinery, 26:618{630, 1979. 17. M. Yannakakis. Node-deletion problems on bipartite graphs. SIAM Journal on Computing, 10:310{ 327, 1981.. REFERENCES 1. A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, M.A., 1974. 2. H.L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209:1{45, 1998. 3. H.L. Bodlaender and T. Kloks. EÆcient and constructive algorithms for the pathwidth and treewidth of graphs. Journal of Algorithms, 21:358{402, 1996. 4. A. Brandstadt, V.B. Le, and J.P. Spinrad. Craph Classes: A Survey. SIAM, 1999. 5. P.W. Goldberg, M.C. Golumbic, H. Kaplan, and R. Shamir. Four strikes against physical mapping of dna. Journal of Computational Biology, 2:139{ 152, 1995. 6. M.C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980. 7. F. Harary. A characterization of block graphs. Canadian Mathematic Bull, 6:1{6, 1982. 8. H. Kaplan, R. Shamir, and R.E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28:1906{ 1922, 1999. 6.

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